Symbolic and Numeric Scientific Computation in Maple K.O. Geddes and H.Q. Le Symbolic Computation Group School of Computer Science University of Waterloo Waterloo ON N2L 3G1 CANADA {kogeddes,hqle}@scg.math.uwaterloo.ca http://www.uwaterloo.ca/~kogeddes Abstract This paper is an exposition of some recent advances in the Maple computer algebra system for both symbolic and numeric scientific computation. Facilities for matrix computations include a convenient syntax for matrix operations, a seamless interface between symbolic and numeric modes, and access to hardware floating point speed for numerical matrices. Techniques for computing exact symbolic solutions of differential equations include decision procedures for special types of DEs, the classification of DEs into known classes, and symmetry-based methods. New numerical solvers which have been incorporated into Maple take full advantage of hardware floating point speed, for definite integrals, IVPs and BVPs. Hybrid symbolic-numeric computational strategies are discussed in the context of evaluating definite integrals in the presence of integrand singularities.

Some of the features appearing in recent versions of Maple [Monagan02a, Monagan02b] for computational linear algebra are highlighted by the following examples. The examples are trivial, chosen simply to illustrate the convenient syntax for matrix operations.

restart; A := < < 1 | -1 | 2 > , < -3 | 4 | -5 > , < 5 | -6 | 6 > >;

LUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMvSSQlaWRHRiciKiEpZltYIg==

b := < 2, 0, -3 >;

LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSdjb2x1bW5HRig2Iy9JJCVpZEdGKCIqVzxdWCI=

A.b;

LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSdjb2x1bW5HRig2Iy9JJCVpZEdGKCIqWyFmaDk=

A^2;

LUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMvSSQlaWRHRiciKndJRVki

A^(-1);

LUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMvSSQlaWRHRiciKkdYTVki

A . A^(-1);

LUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMvSSQlaWRHRiciKk9sVFki

with(LinearAlgebra): Determinant(A);

ISIk

Eigenvalues(A);

LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSdjb2x1bW5HRig2Iy9JJCVpZEdGKCIqa1lMWiI=

x := LinearSolve(A, b);

LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSdjb2x1bW5HRig2Iy9JJCVpZEdGKCIqQ3QncDk=

Norm(b - A.x);

IiIh

Of course, multiplying NiMqJiklIkFHLCQiIiIhIiJGJyUiYkdGJw== yields the solution NiMlInhH . A^(-1).b;

LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSdjb2x1bW5HRig2Iy9JJCVpZEdGKCIqV3UncDk=

(P,L,U) := LUDecomposition(A);

NiUtSSdNYXRyaXhHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2Iy9JJCVpZEdGKCIqJyplJnk5LUYkNiMvRisiKjtmI3k5LUYkNiMvRisiKi8vJHk5

P.L.U;

LUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMvSSQlaWRHRiciKmddIno5

Norm(% - A);

IiIh

Illustrate the solution of NiMvKiYlIkFHIiIiJSJ4R0YmJSJiRw== via LU factorization; i.e., solve NiMvKiolIlBHIiIiJSJMR0YmJSJVR0YmJSJ4R0YmJSJiRw== for NiMlInhH . First solve NiMvKiYlIkxHIiIiJSJ5R0YmKiYpJSJQRyUiVEdGJiUiYkdGJg== for NiMlInlH . y := ForwardSubstitute(L, Transpose(P).b);

LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSdjb2x1bW5HRig2Iy9JJCVpZEdGKCIqd285WyI=

Then solve NiMvKiYlIlVHIiIiJSJ4R0YmJSJ5Rw== for NiMlInhH . x := BackwardSubstitute(U, y);

LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSdjb2x1bW5HRig2Iy9JJCVpZEdGKCIqLyc0Ilsi

Norm(A.x - b);

IiIh

F := RandomMatrix(5,5, generator=-10.0..10.0);

LUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMvSSQlaWRHRiciKktrT1si

G := RandomMatrix(5,3, generator=-10.0..10.0);

LUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMvSSQlaWRHRiciKi9lUVsi

Operations on matrices containing only hardware floating point entries are performed at hardware floating point speed. Some timing information is presented in examples in the following section. F.G;

LUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMvSSQlaWRHRiciKndsU1si

eigs := Eigenvalues(F);

LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSdjb2x1bW5HRig2Iy9JJCVpZEdGKCIqOyUqW1si

poly := CharacteristicPolynomial(F,z);

LC4kIitUSDpsOCEiJSIiIkkiekc2IiQhKygqUlZ4QiEiJiokRiciIiMkIisyJFsxciUhIicqJEYnIiIkJCErN2t3MTUhIigqJEYnIiIlJCErQ0xjNUIhIikqJEYnIiImRiY=

factor(poly);

KigsJkkiekc2IiIiIiQiK2FMJykpWyIhIilGJkYmLCgqJEYkIiIjRiZGJCQhKz1GYCoqUSEiKiQiK2kpeSZSSUYpRiZGJiwoRitGJkYkJCErMU1aNE1GKSQiK1F2YztJISIoRiZGJg==

(Q,R) := QRDecomposition(G): Q;

LUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMvSSQlaWRHRiciKmdbc1si

R;

LUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMvSSQlaWRHRiciKmc/Vlsi

Check that NiMqJiklIlFHJSJURyIiIkYlRic= is the identity matrix. Use Maple's fnormal function to round the entries to 15 digits. map(fnormal, Transpose(Q).Q, 15);

LUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMvSSQlaWRHRiciKiF5U2I5

restart; with(LinearAlgebra): n := 100: Generate a random square matrix of order NiMlIm5H and a random vector of order NiMlIm5H. By right-clicking on the output structure in an interactive Maple session, one can choose from a context menu various operations to be computed or choose to browse the matrix to view its structure. A := RandomMatrix(n,n, generator=-100.0..100.0);

LUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMvSSQlaWRHRiciKm9dN1ki

b := RandomVector(n, generator=-100.0..100.0);

LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSdjb2x1bW5HRig2Iy9JJCVpZEdGKCIqI3pObTk=

Determine the computation time to solve a linear system of order NiMlIm5H in NiMlI2hmRw== (hardware float) mode, for comparison with the time for software float mode. hf_time := time( LinearSolve(A, b) );

JCIkPCMhIiQ=

x := LinearSolve(A, b);

LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSdjb2x1bW5HRig2Iy9JJCVpZEdGKCIqRy5VWiI=

The environment variable UseHardwareFloats controls whether hardware floats or software floats are used for the computation. Determine the computation time to solve the same linear system in software floats. Note that the speedup obtained by using hardware floats is significant. UseHardwareFloats := false: sf_time := time( LinearSolve(A, b) );

JCIlSj8hIiQ=

hf_SpeedUp := evalf[3]( sf_time / hf_time );

JCIkTiohIiM=

UseHardwareFloats := true:

Here we illustrate the following determinant relationship: if NiMvJSJBRyooJSJQRyIiIiUiTEdGJyUiVUdGJw== then NiMvLSUkZGV0RzYjJSJBRyooLUYlNiMlIlBHIiIiLUYlNiMlIkxHRiwtRiU2IyUiVUdGLA== . (P,L,U) := LUDecomposition(A): L;

LUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMvSSQlaWRHRiciKitqISlbIg==

Determinant(L); # We know that the value is 1

JCIjNSEiIg==

U;

LUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMvSSQlaWRHRiciKjM3Q1oi

Determinant(U);

JCIrOD9JJCp6IiRXIw==

NiMtJSRkZXRHNiMlIlVH is simply the product of the diagonal elements: detU := mul(U[i,i], i=1..n);

JCIrOD9JJCp6IiRXIw==

time_detA := time( Determinant(A) );

JCIkcSIhIiQ=

Determinant(A);

JCErOD9JJCp6IiRXIw==

NiMtJSRkZXRHNiMlIkFH is equal to NiMtJSRkZXRHNiMlIlVH up to the sign determined by the permutation matrix, so NiMvLSUkZGV0RzYjJSJQRywkIiIiISIi in this example. Since NiMlIlBH is an exact integer matrix whereas NiMlIkFH is a hardware floating point matrix, the computation time for explicitly computing NiMtJSRkZXRHNiMlIlBH is large compared with the computation time for computing NiMtJSRkZXRHNiMlIkFH . Of course, one does not explicitly form the permutation matrix and compute its determinant in the manner shown here as part of a method for computing NiMtJSRkZXRHNiMlIkFH . The comparison of computation times is shown here to illustrate the speed advantages of hardware floating point mode. P;

LUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMvSSQlaWRHRiciKktLRlgi

time_detP := time( Determinant(P) );

JCIkTSghIiQ=

Determinant(P);

ISIi

TimeRatio := evalf[3]( time_detP / time_detA );

JCIkSyUhIiM=

In this section we illustrate the solution of a linear least squares problem via QR decomposition. The problem being solved is simple and the point is to convey some of the facilities in Maple for problem solving (and for teaching). Consider the problem of calculating a least squares fit by a polynomial to a set of data. For the purposes of this demonstration, create a set of "experimental data" by first generating a random polynomial and then for a chosen list of data points, let the corresponding function values be perturbations of the values of the polynomial at these data points.

Generate a random polynomial NiMtJSJwRzYjJSJ4Rw== of degree NiMsJiUibkciIiJGJSEiIg== (i.e., the number of coefficients is NiMlIm5H ) . restart; with(RandomTools): n := 3: p := unapply( Generate(polynom(integer(range=1..10), x, degree=n-1)), x );

Zio2I0kieEc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCgiIigiIiI5JCIjNSokRiwiIiMiIidGJUYlRiU=

Choose a number, NiMlJU5wdHNH, of data points in a specified range NiM7JSNsb0clI2hpRw== for the independent variable. Define the corresponding function values for the "experimental data" to be the values of NiMtJSJwRzYjJSJ4Rw== at these data points, perturbed by random perturbations in a chosen range NiM7LCQlKGVwc2lsb25HISIiRiU= . Npts := 7: lo := 0.0: hi := 3.0: incr := evalf[2]( (hi-lo)/(Npts-1) ): X := [ seq(lo + (i-1)*incr, i = 1..Npts) ];

NykkIiIhRiQkIiNdISIjJCIkKyJGJyQiJF0iRickIiQrI0YnJCIkXSNGJyQiJCskRic=

epsilon := 0.2;

JCIiIyEiIg==

perturb := Generate(list(float(range=0..2*epsilon, digits=2), Npts)): perturb := map(t -> t-epsilon, perturb);

NykkISIkISIjJCEkSyJGJCQhJFoiRiQkISIiRiUkISRgIkYkJCEkQSJGJCQhJGsiRiQ=

data := [ seq([X[i], p(X[i])*(1+perturb[i])], i=1..Npts) ];

Nyk3JCQiIiFGJSQiJHonISIjNyQkIiNdRigkIiorKz08IiEiKDckJCIkKyJGKCQiKisrPic+Ri43JCQiJF0iRigkIikrXTlOISInNyQkIiQrI0YoJCIqKysoPlZGLjckJCIkXSNGKCQiKisrQDUnRi43JCQiJCskRigkIiorK3dnKEYu

DataPlot := plot(data, x=lo..hi, style=point,symbol=BOX,symbolsize=14): DataPlot;

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

Let the vector NiMlImFH of size NiMlIm5H represent the coefficients of the polynomial fit: NiMvLSUicEc2IyUieEcsKiYlImFHNiMiIiJGLComJkYqNiMiIiNGLEYnRixGLCUmLn4ufi5HRiwqJiZGKjYjJSJuR0YsKUYnLCZGNUYsRiwhIiJGLEYs . In matrix formulation, we must solve an overdetermined linear system NiMvKiYlIkFHIiIiJSJhR0YmJSJiRw== for NiMlImFH . More precisely, we wish to find the vector NiMlImFH which minimizes NiMsJiomJSJBRyIiIiUiYUdGJkYmJSJiRyEiIg== in the least squares norm, where the matrix NiMlIkFH and right-hand-side vector NiMlImJH are formed from the data values as shown below. We apply the QR decomposition of NiMlIkFH to solve this linear least squares problem. with(LinearAlgebra): A := Matrix(Npts, n, (i,j) -> data[i][1]^(j-1));

LUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMvSSQlaWRHRiciKktTclki

b := Vector(Npts, i -> data[i][2]);

LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSdjb2x1bW5HRig2Iy9JJCVpZEdGKCIqP3dyWSI=

(Q,R) := QRDecomposition(A): Q;

LUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMvSSQlaWRHRiciKmtVKHA5

R;

LUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMvSSQlaWRHRiciKjtuLFoi

Since NiMvJSJBRyomJSJRRyIiIiUiUkdGJw== , solve NiMvKiYlIlJHIiIiJSJhR0YmKiYpJSJRRyUiVEdGJiUiYkdGJg== for the vector NiMlImFH . Note that NiMlIlJH is upper triangular. a := BackwardSubstitute(R, Transpose(Q).b);

LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSdjb2x1bW5HRig2Iy9JJCVpZEdGKCIqZy1zWSI=

The desired polynomial fit is as follows, where we choose to round the coefficients to four significant digits. a := map(evalf[4], a): poly := add(a[i]*x^(i-1), i=1..n);

LCgkIiVTZiEiJCIiIkkieEc2IiQiJVI2ISIjKiRGJyIiIyQiJWlTRiU=

Plot both the "experimental data" and the polynomial fit. PolyPlot := plot(poly, x=lo..hi): plots[display]( {DataPlot, PolyPlot} );

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

Of course, given the matrix NiMlIkFH and the vector NiMlImJH, we could have solved the linear least squares problem by directly invoking the LeastSquares function as follows. anew := LeastSquares(A, b): map(evalf[4], anew);

LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSdjb2x1bW5HRig2Iy9JJCVpZEdGKCIqJSl6UFoi

The following command will generate, very quickly, a random sparse matrix with sparsity specified by the density option and with floating point entries in the range specified by the generator option. restart; with(LinearAlgebra): n := 1000: M1 := RandomMatrix(n, n, generator=0.0..1.0, density=0.05, outputoptions=[storage=sparse,datatype=float[8]]);

LUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMvSSQlaWRHRiciKi8uUlki

The following command generates a symmetric, banded matrix. M2 := RandomMatrix(n, n, generator=0.0..1.0, density=0.1, outputoptions=[shape=symmetric,storage=band[0,5],datatype=float[8]]);

LUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMvSSQlaWRHRiciKiF5Sko6

In an interactive Maple session, by right-clicking on the placeholder output seen above one can use the Matrix Browser to view the structure of the matrix. Operations on structured matrices are performed efficiently. Here we multiply matrix NiMlI00yRw== by the vector of size NiMlIm5H with all entries 1.0 . b := Vector(1..n, 1.0): M2.b;

LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSdjb2x1bW5HRig2Iy9JJCVpZEdGKCIqM2FDWSI=

For this example, consider the following banded matrix of order 100. restart; n := 100: B := LinearAlgebra:-RandomMatrix(n, n, generator=0.0..1.0, density=0.05, outputoptions=[shape=symmetric,storage=band[0,5],datatype=float[8]]);

LUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMvSSQlaWRHRiciKjthMFki

By right-clicking on the output NiMlIkJH in an interactive Maple session, the following are some of the operations which can be invoked from the context menu: Browse (to see the band structure) Determinant Rank Singular Values (and Browse the vector of singular values) For example, let us compute the singular values of NiMlIkJH and then calculate the numerical rank [Golub89] of the matrix as follows. singvals := LinearAlgebra:-SingularValues(B);

LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSdjb2x1bW5HRig2Iy9JJCVpZEdGKCIqQys5WSI=

delta := evalhf(DBL_EPSILON) * LinearAlgebra:-Norm(B);

JCIrdClIYDIkISNE

Since the singular values are ordered from largest to smallest, we may apply the following bisection method to find the last "numerically nonzero" singular value. lo := 1: hi := n: while hi-lo > 1 do mid := iquo(lo+hi, 2); if singvals[mid] > delta then lo := mid else hi := mid end if end do: num_rank := lo;

IiNI

By looking at a few singular values near the cutoff, we see that the numerical rank is well-defined in this example. singvals[num_rank-1 .. num_rank+2];

LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSdjb2x1bW5HRig2Iy9JJCVpZEdGKCIqN0wsWSI=

The dsolve command in Maple is a general ODE solver which handles various types of ODE problems including: closed form solutions of a single ODE or a system of ODEs closed form solutions of ODEs with given initial or boundary conditions formal power series solutions of a linear ODE with polynomial coefficients solutions obtained via integral transforms truncated series solutions numerical solutions Some of these types of solutions are illustrated in the following sections.

For the task of computing closed form solutions of a single ODE, two general strategies are employed [ChebTerrab98]: Classification methods determine whether the given ODE matches a recognizable pattern if yes, apply a known solution method for that pattern Special case: Linear ODE, for which decision procedures are implemented Symmetry methods look for generators of the symmetry groups of the given ODE use this information to integrate the ODE, or at least to reduce the order of the ODE

The odeadvisor command is an innovative addition to Maple: especially for teaching and learning classifies a given ODE according to standard textbook classifications provides user access to the classification strategies used by the dsolve command The goal of the odeadvisor command is to classify a given ODE according to standard textbooks [Kamke59, Zwillinger92]. The classification types known to this command are described in the help page ?odeadvisor and they are as follows. First order ODEs Abel, Abel2A, Abel2C, Bernoulli, Chini, Clairaut, dAlembert, exact, homogeneous, homogeneousB, homogeneousC, homogeneousD, homogeneousG, linear, patterns, quadrature, rational, Riccati, separable, sym_implicit Second order ODEs Bessel, Duffing, ellipsoidal, elliptic, Emden, erf, exact_linear, exact_nonlinear, Gegenbauer, Halm, Hermite, Jacobi, Lagerstrom, Laguerre, Lienard, Liouville, linear_ODEs, linear_sym, missing, Painleve, quadrature, reducible, sym_Fx, Titchmarsh, Van_der_Pol High order ODEs quadrature, missing, exact_linear, exact_nonlinear, reducible, linear_ODEs,

restart; with(DEtools): Consider the following first-order nonlinear equation. ode[1] := x*diff(y(x),x)+a*y(x)^2-y(x)+b*x^2;

LCoqJkkieEc2IiIiIi1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJ5R0YlNiNGJEYkRiZGJiomSSJhR0YlRiZGKyIiI0YmRishIiIqJkkiYkdGJUYmRiRGMEYm

Ask for information about the ODE. odeadvisor(ode[1]);

NyU3JEktX2hvbW9nZW5lb3VzRzYiSShjbGFzc35ER0YlSSpfcmF0aW9uYWxHRiVJKV9SaWNjYXRpR0Yl

Solve the ODE. Note: It is not necessary to invoke odeadvisor prior to invoking dsolve. dsolve(ode[1], y(x));

Ly1JInlHNiI2I0kieEdGJSwkKiotSSR0YW5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2IywmKiZGJyIiIiomSSJiR0YlRjJJImFHRiVGMiNGMiIiI0YyKiZJJF9DMUdGJUYyRjNGNkYyRjJGJ0YyRjNGNkY1ISIiRjo=

This is another first-order nonlinear equation. ode[2] := (2*y(x)-x)*diff(y(x),x)-y(x)-2*x;

LCgqJiwmLUkieUc2IjYjSSJ4R0YnIiIjRikhIiIiIiItSSVkaWZmRyUqcHJvdGVjdGVkRzYkRiVGKUYsRixGJUYrRikhIiM=

odeadvisor(ode[2]);

NyY3JEktX2hvbW9nZW5lb3VzRzYiSShjbGFzc35BR0YlSSdfZXhhY3RHRiVJKl9yYXRpb25hbEdGJTclSSZfQWJlbEdGJUkpMm5kfnR5cGVHRiVGJg==

dsolve(ode[2], y(x));

NiQvLUkieUc2IjYjSSJ4R0YmKiYsJiomRigiIiJJJF9DMUdGJkYsI0YsIiIjKiQsJiomRihGL0YtRi8iIiYiIiVGLEYuRi5GLEYtISIiL0YkKiYsJkYrRi5GMCNGNUYvRixGLUY1

Consider the following second-order linear equation. ode[3] := x^2*diff(y(x),x,x)+x*diff(y(x),x)-(x^2+n^2)*y(x);

LCgqJkkieEc2IiIiIy1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtRig2JC1JInlHRiU2I0YkRiRGJCIiIkYwKiZGJEYwRitGMEYwKiYsJiokRiRGJkYwKiRJIm5HRiVGJkYwRjBGLUYwISIi

odeadvisor(ode[3]);

NyM3JEkoX0Jlc3NlbEc2IkkqX21vZGlmaWVkR0Yl

dsolve(ode[3], y(x));

Ly1JInlHNiI2I0kieEdGJSwmKiZJJF9DMUdGJSIiIi1JKEJlc3NlbElHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2JEkibkdGJUYnRitGKyomSSRfQzJHRiVGKy1JKEJlc3NlbEtHRi5GMUYrRis=

The following second-order nonlinear equation can be classified, but the general symbolic solution can only be expressed in an implicit form. ode[4] := diff(y(x),x,x)-mu*(1-y(x)^2)*diff(y(x),x)+y(x) = 0;

LywoLUklZGlmZkclKnByb3RlY3RlZEc2JC1GJTYkLUkieUc2IjYjSSJ4R0YsRi5GLiIiIiooSSNtdUdGLEYvLCZGL0YvKiRGKiIiIyEiIkYvRihGL0Y1RipGLyIiIQ==

odeadvisor(ode[4]);

NyQ3JEkrXzJuZF9vcmRlckc2IkkrX21pc3NpbmdfeEdGJUktX1Zhbl9kZXJfUG9sR0Yl

Numerical solutions of Van der Pol's equation are considered in Example 4.4.

ode[5] := diff(y(x),x) = F((y(x)-x*ln(x))/x) + ln(x);

Ly1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJ5RzYiNiNJInhHRilGKywmLUkiRkdGKTYjKiYsJkYnIiIiKiZGK0YyLUkjbG5HNiRGJUkoX3N5c2xpYkdGKUYqRjIhIiJGMkYrRjhGMkY0RjI=

Lie symmetry methods [ChebTerrab98] are used to express the solution, in this case in an implicit form in terms of Maple's RootOf construct. dsolve(ode[5], y(x));

Ly1JInlHNiI2I0kieEdGJSwmKiYtSSdSb290T2ZHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2IywoLUkjbG5HRixGJiEiIi1JJkludGF0R0YsNiQqJCwoSSNfYUdGLCIiIkY6RjotSSJGR0YlNiNGOUYzRjMvRjlJI19aR0YsRjNJJF9DMUdGJUY6RjpGJ0Y6RjoqJkYnRjpGMUY6Rjo=

For the following particular choice of the function NiMlIkZH , the above ODE is a Riccati equation. F := u -> u^2;

Zio2I0kidUc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlKiQ5JCIiI0YlRiVGJQ==

ode[5];

Ly1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJ5RzYiNiNJInhHRilGKywmKiYsJkYnIiIiKiZGK0YvLUkjbG5HNiRGJUkoX3N5c2xpYkdGKUYqRi8hIiIiIiNGKyEiI0YvRjFGLw==

odeadvisor(ode[5], y(x));

NyQ3JEkrXzFzdF9vcmRlckc2Ikk4X3dpdGhfbGluZWFyX3N5bW1ldHJpZXNHRiVJKV9SaWNjYXRpR0Yl

Request that Lie symmetry methods be used to compute the solution. dsolve(ode[5], y(x), 'Lie');

Ly1JInlHNiI2I0kieEdGJSwkKihGJyIiIiwoKiYiIiYjRioiIiMtSSNsbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJUYmRipGLyokRi1GLkYqLUkldGFuaEdGMjYjLCQqJiwmRjBGKkkkX0MxR0YlRipGKkYtRi5GLiEiJkYqRi1GLiNGKiIjNQ==

Show the symmetry generators (a set of infinitesimals which leave the ODE invariant). symgen(ode[5]);

NyQvSSRfeGlHJSpwcm90ZWN0ZWRHSSJ4RzYiL0klX2V0YUdGJSwmRiYiIiJJInlHRidGKw==

Maple's dsolve can resolve the equivalence between a given ODE and one having 2F1, 1F1 or 0F1 hypergeometric solutions, whenever that equivalence involves power composed with Moebius transformations (see the help page ?dsolve,hypergeometric).

restart; PDEtools[declare](y(x), prime=x);

KigtSSJ5RzYiNiNJInhHRiUiIiJJOXdpbGx+bm93fmJlfmRpc3BsYXllZH5hc0dGJUYoRiRGKA==

KihJPGRlcml2YXRpdmVzfndpdGh+cmVzcGVjdH50b0c2IiIiIkkieEdGJEYlSVpvZn5mdW5jdGlvbnN+b2Z+b25lfnZhcmlhYmxlfndpbGx+bm93fmJlfmRpc3BsYXllZH53aXRofidHRiRGJQ==

ode := diff(y(x),x,x) = 3/16*(x^2+10)*(4*x^6+16*x^4+23*x^2-10)/(x^2+2)^2/(x^2+5)^2/x^2*y(x);

Ly1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtRiQ2JC1JInlHNiI2I0kieEdGK0YtRi0sJCouLCYqJEYtIiIjIiIiIiM1RjNGMywqKiRGLSIiJyIiJSokRi1GOCIjO0YxIiNCISM1RjNGMywmRjFGM0YyRjMhIiMsJkYxRjMiIiZGM0Y+Ri1GPkYpRjMjIiIkRjo=

sol := dsolve(ode);

Ly1JInlHNiI2I0kieEdGJSwmKixJJF9DMUdGJSIiIiwmKiRGJyIiI0YrIiImRisjRisiIiktSSpoeXBlcmdlb21HSShfc3lzbGliR0YlNiU3JCNGKyIiJSNGK0YuNyMjIiIkRjgsJComRidGLiwmRi1GLiIjNUYrISIiISIkRitGJ0Y3LCZGLUYrRi5GK0Y5RisqLEkkX0MyR0YlRitGLCNGQUYxLUYzNiU3JEY5Rjs3IyNGL0Y4Rj1GK0YnRjtGQ0Y5Ris=

ode := diff(y(x),x,x) = 27*x*y(x)/(2*x+1)/(x-1)^4;

Ly1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtRiQ2JC1JInlHNiI2I0kieEdGK0YtRi0sJCoqRi0iIiJGKUYwLCZGMEYwRi0iIiMhIiIsJkYtRjBGM0YwISIlIiNG

sol := dsolve(ode);

Ly1JInlHNiI2I0kieEdGJSwmKihJJF9DMUdGJSIiIiwmRidGKyEiIkYrRistSStXaGl0dGFrZXJNRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiUjRi0iIiMjRitGNSomLCZGNUYrRiciIiVGK0YsRi1GK0YrKihJJF9DMkdGJUYrRixGKy1JK1doaXR0YWtlcldHRjBGM0YrRis=

convert(sol, StandardFunctions);

Ly1JInlHNiI2I0kieEdGJSwmKihJJF9DMUdGJSIiIiwmRidGKyEiIkYrRistSStXaGl0dGFrZXJNRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiUjRi0iIiMjRitGNSomLCZGNUYrRiciIiVGK0YsRi1GK0YrKihJJF9DMkdGJUYrRixGKy1JK1doaXR0YWtlcldHRjBGM0YrRis=

ode := diff(y(x),x,x) = 1/20/x^2*(405*x^6-5670*x^4+58604*x^2+13720)/(3*x^2-14)^3*y(x);

Ly1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtRiQ2JC1JInlHNiI2I0kieEdGK0YtRi0sJCoqRi0hIiMsKiokRi0iIiciJDAlKiRGLSIiJSElcWMqJEYtIiIjIiYvJ2UiJj9QIiIiIkY8LCZGOCIiJCEjOUY8ISIkRilGPCNGPCIjPw==

sol := dsolve(ode);

Ly1JInlHNiI2I0kieEdGJSwmKihJJF9DMUdGJSIiIiwmKiRGJyIiJEYuRichIzkjRisiIiMtSSpoeXBlcmdlb21HSShfc3lzbGliR0YlNiU3IjcjRissJComRidGMSwmKiRGJ0YxIiM6ISNxRishIiIiIzlGK0YrKihJJF9DMkdGJUYrRixGMC1JKEJlc3NlbEtHNiQlKnByb3RlY3RlZEdGNDYkIiIhLCQqJiomRidGMSwmRjtGLkYvRitGPkYwIiNxRjAjRjEiIiZGK0Yr

convert(sol, Bessel);

Ly1JInlHNiI2I0kieEdGJSwmKihJJF9DMUdGJSIiIiwmKiRGJyIiJEYuRichIzkjRisiIiMtSShCZXNzZWxKRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiQiIiEsJComLCYqJEYnRjEhJDUjIiQhKSpGKyMhIiJGMUYnRisiI0dGK0YrKihJJF9DMkdGJUYrRixGMC1JKEJlc3NlbEtHRjQ2JEY4LCQqJiomRidGMSwmRjxGLkYvRitGQEYwIiNxRjAjRjEiIiZGK0Yr

An example with fractional power coefficients, whose solution can be expressed in elementary form. ode := diff(y(x),x,x) = 1/4*(2/3*1/(x^(5/6))+2/x^(2/3))/x*y(x) + 1/2*(-1+x^(1/6))/x*diff(y(x),x);

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sol := dsolve(ode);

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convert(sol, elementary);

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There are algorithms in Maple to compute various types of formal power series solutions [Abramov00] for a linear ODE with polynomial coefficients. The following examples will serve to illustrate.

restart; ode := (3*x^2-6*x+3)*diff(y(x),x,x) + (12*x-12)*diff(y(x),x) + 6*y(x);

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dsolve(ode, y(x), formal_series, coeffs=polynomial);

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ode := 2*x*(x-1)*diff(y(x),x,x) + (7*x-3)*diff(y(x),x) + 2*y(x) = 0;

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dsolve(ode, y(x), type=formal_series, coeffs=hypergeom);

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ode := diff(y(x),x,x) + (x-1)*y(x);

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dsolve(ode, y(x), type=formal_series, coeffs=mhypergeom);

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The DEtools package provides user tools for manipulating ODEs including: functions for visualization e.g. DEplot, dfieldplot, phaseportrait general manipulation of differential equations e.g. Dchangevar, convertsys, indicialeq, reduceOrder functions for constructing closed form solutions e.g. constcoeffsols, exactsol, expsols, linearsol, ratsols, separablesol functions for differential operators e.g. DFactor, adjoint, formal_sol Some examples using the functions for visualization are presented below.

restart; with(DEtools): de1 := cos(x)*diff(y(x),x$3)-diff(y(x),x$2)+Pi*diff(y(x),x)=y(x)-x;

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DEplot(de1, y(x), x=-2.5..1.4, [[y(0)=1,D(y)(0)=2,(D@@2)(y)(0)=1]], y=-4..5, stepsize=.05);

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de2a := diff(x(t),t) = x(t)*(1-y(t)); de2b := diff(y(t),t) = .3*y(t)*(x(t)-1);

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Ly1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJ5RzYiNiNJInRHRilGKywkKiZGJyIiIiwmLUkieEdGKUYqRi4hIiJGLkYuJCIiJEYy

dfieldplot([de2a,de2b], [x(t),y(t)], t=-2..2, x=-1..2, y=-1..2, arrows=LARGE, title="Lotka-Volterra model", color=[.3*y(t)*(x(t)-1),x(t)*(1-y(t)),.1]);

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de3 := diff(y(x),x) = 1/2*(-x-(x^2+4*y(x))^(1/2));

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dfieldplot(de3, y(x), x=-3..3, y=-3..2, title="Restricted domain", color=1/2*(-x-(x^2+4*y)));

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de4a := D(x)(t) = y(t)-z(t); de4b := D(y)(t) = z(t)-x(t); de4c := D(z)(t) = x(t)-y(t)*2;

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Ly0tSSJERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNJInpHRik2I0kidEdGKSwmLUkieEdGKUYsIiIiLUkieUdGKUYsISIj

phaseportrait([de4a,de4b,de4c], [x(t),y(t),z(t)], t=-2..2, [[x(0)=1,y(0)=0,z(0)=2]], stepsize=.05, scene=[z(t),x(t)], linecolour=sin(t*Pi/2), method=classical[foreuler]);

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de5 := D(y)(x) = -y(x)-x^2;

Ly0tSSJERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiNJInlHRik2I0kieEdGKSwmLUYrRiwhIiIqJEYtIiIjRjA=

phaseportrait(de5, y(x), x=-1..2.5, [[y(0)=0],[y(0)=1],[y(0)=-1]], title="Asymptotic solution",colour=magenta, linecolor=[gold,yellow,wheat]);

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The Maple scientific computation environment supports both symbolic and numeric mathematical computation, as previous sections have illustrated. In this section a new paradigm, hybrid symbolic-numeric computation, is employed to achieve an enhanced level of computational power. The overall strategy is to apply an appropriate combination of symbolic mathematical analysis and numerical computation. The hybrid symbolic-numeric methods illustrated here are part of an automated numerical integration strategy [Geddes86, Geddes92] implemented in Maple for the numerical evaluation of definite integrals. In particular, symbolic analysis techniques are used to deal with integrand singularities.

restart; f := x^2*ln(x)*exp(-x^2);

KihJInhHNiIiIiMtSSNsbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRiMiIiItSSRleHBHRig2IywkKiRGI0YlISIiRiw=

plot( f, x=0..4 );

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

Form the integral onNiMtJSFHNiQiIiEiIiU= and attempt to evaluate in symbolic mode. intf := Int( f, x=0..4 ): value( intf );

LUkkaW50RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQqKEkieEdGJyIiIy1JI2xuR0YkNiNGKiIiIi1JJGV4cEdGJDYjLCQqJEYqRishIiJGLy9GKjsiIiEiIiU=

The integral does not evaluate in symbolic mode. Apply direct numerical integration (at standard precision). evalf( intf ); # Invokes compiled NAG routines.

JCIrXTNGJTMpISM3

Or at higher precision. evalf[25]( intf );

JCI6Q2RKVXRHSCpcM0YlMykhI0Y=

Notice from the graph that the integral onNiMtJSFHNiQiIiElKWluZmluaXR5Rw== will have approximately the same numerical value. Curiously, for the infinite integral a closed form expression can be obtained! newintf := Int( f, x=0..infinity ): newintf = value( newintf );

Ly1JJEludEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkKihJInhHRigiIiMtSSNsbkdGJTYjRisiIiItSSRleHBHRiU2IywkKiRGK0YsISIiRjAvRis7IiIhSSlpbmZpbml0eUdGJiwoKiRJI1BpR0YmI0YwRiwjRjAiIiUqJkY9Rj5JJmdhbW1hR0YmRjAjRjYiIikqJkY9Rj4tRi42I0YsRjAjRjZGQA==

Of course, this symbolic formula can be evaluated numerically. evalf( rhs(%) );

JCIpJCpmJTMpISM1

Or, we may apply direct numerical integration to the infinite integral. evalf( newintf ); # Invokes compiled NAG routines.

JCIraSQqZiUzKSEjNw==

Note: Numerical evaluation of the exact formula loses some precision. The result from numerical integration is correct to 10 significant digits.

g := sin(x)*ln(x)*exp(-x^3);

KigtSSRzaW5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2I0kieEdGKCIiIi1JI2xuR0YlRilGKy1JJGV4cEdGJTYjLCQqJEYqIiIkISIiRis=

plot( g, x=0..3 );

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

Form the integral onNiMtJSFHNiQiIiEiIiQ=, attempt symbolic evaluation, and apply numerical integration. intg := Int( g, x=0..3 ): value( intg );

LUkkaW50RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQqKC1JJHNpbkdGJDYjSSJ4R0YnIiIiLUkjbG5HRiRGLEYuLUkkZXhwR0YkNiMsJCokRi0iIiQhIiJGLi9GLTsiIiFGNg==

evalf( intg );

JCErZV4peSY+ISM1

Evaluating numerically to 25 digits yields the following result. evalf[25]( intg );

JCE6K0tLbHMyKVtlXil5Jj4hI0Q=

Note that the integrand has a logarithmic singularity at x=0 as the following series expansion shows. series( g, x=0 );

Ky1JInhHNiItSSNsbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjRiMiIiIsJEYlIyEiIiIiJyIiJCwkRiVGLiIiJSwkRiUjRisiJD8iIiImLUkiT0dGKDYjRitGLw==

The hybrid symbolic-numeric technique includes the concept of term-by-term integration of such a series expansion.

restart; h := ln(1 - cos(2*x)): Int(h, x=0..1);

LUkkSW50RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQtSSNsbkdGJDYjLCYiIiJGLS1JJGNvc0dGJDYjLCRJInhHRiciIiMhIiIvRjI7IiIhRi0=

Note that the graph of this integrand goes to NiMsJCUpaW5maW5pdHlHISIi at NiMvJSJ4RyIiIQ==. However, this is an integrable singularity - it behaves like NiMtJSNsbkc2IyUieEc= near NiMvJSJ4RyIiIQ==. plot(h, x=0..1);

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

Note: The technique of subtracting off a singularity illustrated below, takes place automatically within Maple's numerical integration routines. Suppose it is desired to compute the result to 25 digits of accuracy. The generalized series expansion of NiMlImhH at NiMvJSJ4RyIiIQ== takes the following form. Digits := 25: series(h, x=0, 8);

Ky1JInhHNiIsJi1JI2xuRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMiIiMiIiItRic2I0YjRiwiIiEjISIiIiIkRiwjRjIiIyEqIiIlIyEiIyIlTkciIictSSJPR0YpNiNGLSIiKQ==

The non-regular part is q := 2*ln(x);

LCQtSSNsbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSJ4R0YoIiIj

The new expression newh := h - q;

LCYtSSNsbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCYiIiJGKy1JJGNvc0dGJTYjLCRJInhHRigiIiMhIiJGKy1GJDYjRjAhIiM=

is analytic on the interval [0, 1]. Thus it can be integrated easily by the default numerical integration method. r1 := evalf(Int(newh, x=0..1));

JCI6J0hIOlZheHcmb25xeiYhI0Q=

Integrating NiMlInFH is easy because it has the indefinite integral int(q, x);

LCYqJkkieEc2IiIiIi1JI2xuRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiNGJEYmIiIjRiQhIiM=

and its definite integral can therefore be computed symbolically. r2 := int(q, x=0..1);

ISIj

Finally, summing the two values, we obtain the value for the original definite integration problem. Int(h, x=0..1) = r1 + r2;

Ly1JJEludEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkLUkjbG5HRiU2IywmIiIiRi4tSSRjb3NHRiU2IywkSSJ4R0YoIiIjISIiL0YzOyIiIUYuJCE6cXElb2JDS1VKS0g/OSEjQw==

F := sqrt(sin(x)): Int(F, x=0..2);

LUkkSW50RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQqJC1JJHNpbkdGJDYjSSJ4R0YnIyIiIiIiIy9GLTsiIiFGMA==

Note: The change of variables illustrated below takes place automatically within Maple's numerical integration routines. The generalized series expansion of the integrand NiMlIkZH at NiMvJSJ4RyIiIQ== is of the form series(F, x=0, 5);

LCgqJEkieEc2IiMiIiIiIiNGJyokRiQjIiImRigjISIiIiM3LUkiT0clKnByb3RlY3RlZEc2IyokRiQjIiIqRihGJw==

Applying the change of variables NiMvJSJ0Ry0lJXNxcnRHNiMlInhH yields r3 := Int( eval(F, x=t^2) * diff(t^2, t), t = 0..sqrt(2) );

LUkkSW50RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQsJComLUkkc2luR0YkNiMqJEkidEdGJyIiIyMiIiJGMEYvRjJGMC9GLzsiIiEqJEYwRjE=

The new integrand is analytic on the interval of integration. Therefore it can be integrated easily by the default numerical integration method, even at high accuracy. Suppose that the result is desired to 50 digits of accuracy. evalf[50](r3);

JCJTI2Z4PVA2QU9OJGYiemc1JHpdXW0qPiQzTXM/OyEjXA==

restart; G := exp(v - v^2/2) / (1 + 1/2*exp(v)): Int(G, v=0..infinity);

LUkkSW50RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQqJi1JJGV4cEdGJDYjLCZJInZHRiciIiIqJEYuIiIjIyEiIkYxRi8sJkYvRi8tRis2I0YuI0YvRjFGMy9GLjsiIiFJKWluZmluaXR5R0Yl

Note: The development of the integrand in a generalized series expansion and its integration term-by-term as illustrated below takes place automatically within Maple's numerical integration routines. Suppose it is desired to compute the integral to 20 digits of accuracy. First, the interval of integration is split into NiM7IiIhIiIi and NiM7IiIiJSlpbmZpbml0eUc= . For the finite interval, the default numerical integration method has no difficulty. Digits := 20: r01 := evalf(Int(G, v=0..1));

JCI1IXltQz0hSFtjIWUoISM/

For the infinite interval, the change of variables NiMvJSJ2RyomIiIiRiYlInhHISIi transforms the problem into the following new integration problem. r1inf := Int( -eval(G, v=1/x) * diff(1/x, x), x = 0..1 );

LUkkSW50RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQqKC1JJGV4cEdGJDYjLCYqJEkieEdGJyEiIiIiIiokRi8hIiMjRjAiIiNGMSwmRjFGMS1GKzYjRi4jRjFGNUYwRi9GMy9GLzsiIiFGMQ==

Let the integrand appearing here be named NiMlImdH . The first few terms of the generalized series expansion of NiMlImdH are as follows. g := op(1, r1inf): n := 6: s := `evalf/int/genseries`(g, x, n);

LDAqJi1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjLCQqJEkieEdGKSEiIyEiIiMiIiIiIiNGLUYuRjIqKEYkRjBGLUYuLUYlNiMsJCokRi1GL0YvRjEhIiUqKEYkRjBGLUYuRjRGMiIiKSooRiRGMEYtRi5GNCIiJCEjOyooRiRGMEYtRi5GNCIiJSIjSyooRiRGMEYtRi5GNCIiJiEjay1JIk9HRic2IyokRjQiIidGMQ==

Note that NiMlInNH is a series in increasing powers of NiMtJSRleHBHNiMsJComIiIiRiglInhHISIiRio=. Maple's symbolic integrator determines that the first two terms of NiMlInNH have the following indefinite integrals. terms := [seq(simplify(op(i,s),symbolic), i=1..n)]: int(terms[1], x);

LCQqKEkjUGlHJSpwcm90ZWN0ZWRHIyIiIiIiI0YoRiYtSSRlcmZHNiRGJUkoX3N5c2xpYkc2IjYjLCQqJkYoRiZJInhHRi0hIiJGJkYnRjI=

int(terms[2], x);

LCQqKkkjUGlHJSpwcm90ZWN0ZWRHIyIiIiIiIy1JJGV4cEc2JEYlSShfc3lzbGliRzYiNiNGJkYnRihGJi1JJGVyZkdGKzYjLCYqJkYoRiZJInhHRi0hIiJGJiokRihGJkYmRidGKA==

Similarly, the integral of each term can be computed symbolically. If we compute the definite integral over [0, 0.25] of the successive terms of the generalized series expansion of NiMlImdH , we find that the successive values are as follows. For the numerical evaluation of the symbolic formula for each term, and for the addition of the terms to get an accurate floating point result, it is generally necessary to set the working precision higher for this stage of the computation. Digits := 2*Digits: seq( int(terms[i], x=0..0.25), i = 1..n ): evalf( [ % ] );

NygkIklIT1BHLmxFT3FCXjIiXEphZ2d4ZSIhI1YkIUllMCVvJVtxKXosYl0qb29YRGJkaFFaISNYJCJHSzpDU2tJcj80cjIob2leKGUmPVkiISNXJCFDX1Q0Kj1zJXozOTcpKSlRIyo+L2klISNVJCJAKnlSVSgpKSlIXmYoKmZIclsoWyIhI1MkITwoM1NKI29rIzN3cFdLZVshI1E=

Clearly, the series representation is converging reasonably quickly on this interval (the magnitude of successive terms is decreasing at a good rate). To obtain a numerical result accurate to 20 digits, it is necessary to sum more than six terms of the series as can be seen from the magnitude of the terms above. For this example, 20 terms is more than necessary. n := 20: s := `evalf/int/genseries`(g, x, n): terms := [seq(simplify(op(i,s),symbolic), i=1..n)]: The following loop adds successive terms until the magnitude of the terms becomes negligible for 20 digit accuracy. Digits := 2*Digits: epsilon := 1e-20: t := evalf( int(terms[1], x=0..0.25) ): val := t: for i from 2 to n while abs(t/val) > epsilon do t := evalf( int(terms[i], x=0..0.25) ); val := val + t end do: number_of_terms := i-1;

IiM6

We now have the following result for the integral of NiMlImdH over [0, 0.25] . Digits := 20: r1 := evalf(val);

JCI1aiVmQDcrYSJ6VDohI0I=

For the remaining interval [0.25, 1] , ordinary numerical integration methods encounter no difficulties because there are no nearby singularities. r2 := evalf(Int(g, x=0.25..1));

JCI1Lncsb2lxQjF0YSEjPw==

Add these two values to get the value of the integral NiMlJnIxaW5mRw== . r1inf := r1 + r2;

JCI1aShSIm87aVRndWEhIz8=

Finally, we have obtained the answer for the original integration problem. Int(G, v=0..infinity) = r01 + r1inf;

Ly1JJEludEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkKiYtSSRleHBHRiU2IywmSSJ2R0YoIiIiKiRGLyIiIyMhIiJGMkYwLCZGMEYwLUYsNiNGLyNGMEYyRjQvRi87IiIhSSlpbmZpbml0eUdGJiQiNWExMSY9Iioqb14wOCEjPg==

The most significant improvement in numerical integration capabilities for Maple 8 is the addition of numerical methods for multiple integration. In previous releases, one could form a multiple integral using nested Int expressions, and then invoke the evalf command on the multiple integration problem. The only solution method was to invoke, recursively, one-dimensional numerical integration methods. This was an inefficient approach to numerical multiple integration problems which would succeed only on the simplest problems. In Maple 8, compiled C routines which implement numerical multiple integration methods are automatically invoked for such problems whenever the desired precision is in hardware floating point range (typically about 15 decimal digits).

A numerical multiple integration problem can be specified in a natural way using nested one-dimensional integrals, for example: evalf( Int(...(Int(Int(f, x1=a1..b1), x2=a2..b2), ...), xn=an..bn) ) where the integrand NiMlImZH depends on NiYlI3gxRyUjeDJHJSQuLi5HJSN4bkc= . Such a problem can also be specified using the following special multiple integration notation with a list as the second argument: evalf( Int(f, [x1=a1..b1, x2=a2..b2, ..., xn=an..bn]) . Additional optional arguments can be stated just as in the case of one-dimensional integration. Also as in one-dimensional integration, the integrand NiMlImZH can be specified as a procedure in which case the second argument must be a list of ranges: [a1..b1, a2..b2, ..., an..bn] . Whether a multiple integration problem is stated using nested integrals or using the list notation, the arguments are extracted so as to invoke the same numerical multiple integration routines. See the help page ?evalf/int for further details and examples.

Multiple integrals may be expressed as nested one-dimensional integrals. restart; Int(Int(Int(exp(x+y+z)/((5*x+1)*(10*y+2)*(15*z+3)), x=0..4), y=0..3), z=0..sqrt(2));

LUkkSW50RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQtRiM2JC1GIzYkKiotSSRleHBHRiQ2IywoSSJ4R0YnIiIiSSJ5R0YnRjNJInpHRidGM0YzLCZGMiIiJkYzRjMhIiIsJkY0IiM1IiIjRjNGOCwmRjUiIzoiIiRGM0Y4L0YyOyIiISIiJS9GNDtGQUY+L0Y1O0ZBKiRGOyNGM0Y7

evalf(%);

JCIrRDhoSiQqISM1

Numerical multiple integration may also be invoked using a list syntax. d := (1 - w^2*x^2*y^2*z^2): g := d * cos(w*x*y*z) - d * w*x*y*z * sin(w*x*y*z);

LCYqJiwmIiIiRiUqKkkid0c2IiIiI0kieEdGKEYpSSJ5R0YoRilJInpHRihGKSEiIkYlLUkkY29zRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YoNiMqKkYnRiVGKkYlRitGJUYsRiVGJUYlKi5GJEYlRidGJUYqRiVGK0YlRixGJS1JJHNpbkdGMEYzRiVGLQ==

DesiredIntegral := Int(Int(Int(Int(g, w=0..1), x=0..1), y=0..1), z=0..1);

LUkkSW50RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQtRiM2JC1GIzYkLUYjNiQsJiomLCYiIiJGMioqSSJ3R0YnIiIjSSJ4R0YnRjVJInlHRidGNUkiekdGJ0Y1ISIiRjItSSRjb3NHRiQ2IyoqRjRGMkY2RjJGN0YyRjhGMkYyRjIqLkYxRjJGNEYyRjZGMkY3RjJGOEYyLUkkc2luR0YkRjxGMkY5L0Y0OyIiIUYyL0Y2RkIvRjdGQi9GOEZC

Here we use a list syntax to give the integration command, which would avoid the need to form the nested integral above. evalf(Int(g, [w=0..1, x=0..1, y=0..1, z=0..1]));

JCIreCIpejwoKiEjNQ==

Equivalently, we could have requested numerical evaluation of the nested integral formed above. evalf(DesiredIntegral);

JCIreCIpejwoKiEjNQ==

When low accuracy is sufficient, the Monte Carlo method may be used. h := 1/(2 + sin(Pi*sqrt(87)*(x1+x2+x3+x4+x5+x6)));

KiQsJiIiIyIiIi1JJHNpbkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjKihJI1BpR0YpRiUiIygpI0YlRiQsLkkjeDFHRitGJUkjeDJHRitGJUkjeDNHRitGJUkjeDRHRitGJUkjeDVHRitGJUkjeDZHRitGJUYlRiUhIiI=

evalf(Int(h, [x1=-1..1, x2=-1..1, x3=-1..1, x4=-1..1, x5=-1..1, x6=-1..1], method=_MonteCarlo, epsilon=0.5e-2)): Only trust about 3 digits when epsilon=0.5e-2 . evalf[3](%);

JCIkcCQhIiI=

One may consult the help page ?dsolve,numeric for details about the numerical ODE solvers available in Maple. The examples below illustrate some of the capabilities for numerically solving initial-value problems.

Consider the following Riccati equation NiMlI2RlRw== with initial condition NiMlI2ljRw== . restart; de := diff(y(x),x) = (y(x) - x*ln(x))^2/x^2 + ln(x);

Ly1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJ5RzYiNiNJInhHRilGKywmKiYsJkYnIiIiKiZGK0YvLUkjbG5HNiRGJUkoX3N5c2xpYkdGKUYqRi8hIiIiIiNGKyEiI0YvRjFGLw==

ic := y(1)=1/2;

Ly1JInlHNiI2IyIiIiNGJyIiIw==

For this example, it is possible to compute an exact symbolic solution. soln := dsolve({de,ic}, y(x));

Ly1JInlHNiI2I0kieEdGJSwkKihGJyIiIiwoKiQiIiYjRioiIiMhIiIqJi1JI2xuRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlRiZGKkYtRi4hIiMtSSV0YW5oR0Y0NiMsJEYxRi5GLUYqRi1GLiNGMCIjNQ==

y_exact := unapply(rhs(soln), x): Compute a solution using the default numerical method which is a Runge-Kutta Fehlberg method rkf45. y_numeric := dsolve({de,ic}, type=numeric, range=1..4);

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

Look at the solution at some specific points. y_numeric(1.5);

NyQvSSJ4RzYiJCIjOiEiIi8tSSJ5R0YlNiNGJCQiM29nZnpyPTRqayEjPQ==

y_numeric(4.0);

NyQvSSJ4RzYiJCIjUyEiIi8tSSJ5R0YlNiNGJCQiMyEqeVssJ1IkcGVNISM8

Comparing with the exact answer at NiMvJSJ4RyIiJQ==, we see that the default numerical solution is accurate to about 5 digits. evalf[15]( y_exact(4) );

JCIwX3hTVWcnZU0hIzk=

The numerical solution can be plotted efficiently because all necessary data has already been computed and stored in the procedure NiMlKnlfbnVtZXJpY0c= . plots[odeplot](y_numeric);

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

restart; de := diff(x(t),t,t) = g/l*sin(x(t));

Ly1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtRiQ2JC1JInhHNiI2I0kidEdGK0YtRi0qKEkiZ0dGKyIiIkkibEdGKyEiIi1JJHNpbkc2JEYlSShfc3lzbGliR0YrNiNGKUYw

g := 32; l := 2;

IiNL

IiIj

inits := x(0)=0, D(x)(0)=1/10;

NiQvLUkieEc2IjYjIiIhRigvLS1JIkRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiY2I0YlRicjIiIiIiM1

For this example, the symbolic solution is expressed in a somewhat complicated implicit form which involves the RootOf construct and an unevaluated integral. soln := dsolve({de,inits}, x(t));

Ly1JInhHNiI2I0kidEdGJS1JJ1Jvb3RPZkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjLCYtSSRJbnRHRio2JCwkKiQsJi1JJGNvc0dGKjYjSSNfYUdGKiElK0siJSxLIiIiIyEiIiIiIyIjNS9GODsiIiFJI19aR0YqRj1GJ0Y7

x_exact := unapply(rhs(soln), t): Computing numerical values of NiMlKHhfZXhhY3RH is quite inefficient because fsolve and evalf/int are repeatedly invoked! for tval from 0.0 by 0.2 to 0.6 do 't' = tval, 'x_exact' = evalf( x_exact(tval) ) end do;

NiQvSSJ0RzYiJCIiIUYnL0koeF9leGFjdEdGJUYm

NiQvSSJ0RzYiJCIiIyEiIi9JKHhfZXhhY3RHRiUkIis3KmYtQSMhIzY=

NiQvSSJ0RzYiJCIiJSEiIi9JKHhfZXhhY3RHRiUkIitjVWxRZiEjNg==

NiQvSSJ0RzYiJCIiJyEiIi9JKHhfZXhhY3RHRiUkIishei1oTyIhIzU=

In this case, a numerical solution can be computed much more efficiently. x_numeric := dsolve({de,inits}, type=numeric, range=0..4);

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

for tval from 0.0 by 0.2 to 0.6 do 't' = tval, 'x_numeric' = eval(x(t), x_numeric(tval)) end do;

NiQvSSJ0RzYiJCIiIUYnL0kqeF9udW1lcmljR0YlRiY=

NiQvSSJ0RzYiJCIiIyEiIi9JKnhfbnVtZXJpY0dGJSQiM0V4JD11KCpmLUEjISM+

NiQvSSJ0RzYiJCIiJSEiIi9JKnhfbnVtZXJpY0dGJSQiMz8lcDc6a1snUWYhIz4=

NiQvSSJ0RzYiJCIiJyEiIi9JKnhfbnVtZXJpY0dGJSQiMyFHSVhaOyxoTyIhIz0=

Plot the numerical solution. plots[odeplot](x_numeric);

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

The result computed by the numerical solver rkf45 is a piecewise-polynomial interpolant. It is the interpolant which gets plotted by odeplot. For instructional purposes, we can request the Maple solver to return the piecewise-polynomial solution as the output. Here we restrict to a smaller range so that the output is not too large. x_pwpoly := dsolve({de,inits}, type=numeric, output=piecewise, range=0..0.2): Extract the NiMtJSJ4RzYjJSJ0Rw== solution and display it using 5-digit format. x_pwpoly := eval(x(t), x_pwpoly): evalf[5](x_pwpoly);

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

Compare values of NiMlKXhfcHdwb2x5Rw== with NiMlKHhfZXhhY3RH . for tval from 0.0 by 0.1 to 0.2 do 't' = tval, 'x_pwpoly' = eval(x_pwpoly, t=tval), 'x_exact' = evalf( x_exact(tval) ) end do;

NiUvSSJ0RzYiJCIiIUYnL0kpeF9wd3BvbHlHRiUkIiNJISM8L0koeF9leGFjdEdGJUYm

NiUvSSJ0RzYiJCIiIiEiIi9JKXhfcHdwb2x5R0YlJCIrUTIpby0iISM2L0koeF9leGFjdEdGJSQiK3YxKW8tIkYt

NiUvSSJ0RzYiJCIiIyEiIi9JKXhfcHdwb2x5R0YlJCIrTyRmLUEjISM2L0koeF9leGFjdEdGJSQiKzcqZi1BI0Yt

Compute the piecewise-polynomial solution for range=0..4 and plot it. Of course, it is the same plot as previously displayed. x_pwpoly := dsolve({de,inits}, type=numeric, output=piecewise, range=0..4): x_pwpoly := eval(x(t), x_pwpoly): plot(x_pwpoly, t=0..4);

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

The Van der Pol equation in relaxation oscillation is a famous example of a stiff nonlinear problem. restart; mu := 1000: de := diff(y(x),x,x)-mu*(1-y(x)^2)*diff(y(x),x)+y(x) = 0;

LywoLUklZGlmZkclKnByb3RlY3RlZEc2JC1GJTYkLUkieUc2IjYjSSJ4R0YsRi5GLiIiIiomLCZGL0YvKiRGKiIiIyEiIkYvRihGLyElKzVGKkYvIiIh

inits[1] := y(0)=2, D(y)(0)=0;

NiQvLUkieUc2IjYjIiIhIiIjLy0tSSJERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YmNiNGJUYnRig=

The following attempt to compute a numerical solution by the default non-stiff method leads to trouble. soln[0] := dsolve({de,inits[1]}, type=numeric, range=0..3000):

Warning, cannot evaluate the solution further right of 6.1339278, maxfun limit exceeded (see ?dsolve,maxfun for details)

Specify the option stiff=true so that the default stiff method will be used, namely an implicit Rosenbrock third-fourth order Runge-Kutta method. soln[1] := dsolve({de,inits[1]}, type=numeric, range=0..3000, stiff=true);

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

A plot of the solution shows that there are regions of sharp change. The IVP is stiff where the solution is slowly varying. plots[odeplot](soln[1], [x, y(x)]);

NiQtSSdDVVJWRVNHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JFgsSSlhbnl0aGluZ0clKnByb3RlY3RlZEc2IjYiW2dsJyElIiEhI19xIlMiIzAwMDAwMDAwMDAwMDAwMDA0MDAwMDAwMDAwMDAwMDAwNDA0RTlDQkMxNEU1RTA3MzNGRkY1NURENDIxMTg4NkU0MDVFOUNCQzE0RTVFMThEM0ZGRUE1NTczRDVERUYzRTQwNjZGNThEMEZBQzY3Q0EzRkZERUQ5MkNCMDg2MjkxNDA2RTlDQkMxNEU1RTAyRDNGRkQyRDdGOEQ3QjQ2OEY0MDczMjFGNThEMEZBQzQ4M0ZGQzYzQzI1QjM3MjBDRjQwNzZGNThEMEZBQzY4N0EzRkZCOEU5MjZFNDZDOUM2NDA3QUM5MjQ5MjQ5MjRBQjNGRkFBQjgyMzhGMjBDOTY0MDdFOUNCQzE0RTVFMDJEM0ZGOUI3MUM3REFBN0FFMDQwODEzODI5Q0JDMTRFMkYzRkY4QUMyRDI3NTQ2QzE2NDA4MzIxRjU4RDBGQUM0ODNGRjc4MjFENjJGQjBBOUE0MDg1MEJDMTRFNUUwQTYxM0ZGNjI4QzdDRjJGRUJEMDQwODZGNThEMEZBQzY4N0EzRkY0Nzk2NEMzMzlFQkQxNDA4OERGNThEMEZBQzYzQjNGRjFCMzkyMDk2MEFBQkQ0MDhBQzkyNDkyNDkyNDUzQkZGRjc1QUI2M0IxNkM0QTQwOENCMkYwNTM5NzgyNkNCRkZFQzY2MUJDQjM1Rjg5NDA4RTlDQkMxNEU1RTA4NUJGRkUxMDA1OUI2MzAxN0E0MDkwNDM0M0VCMUExRjIzQkZGRDUxOTFBNkM1RERDMDQwOTEzODI5Q0JDMTRERDdCRkZDODlCQUM1MUU5RDUyNDA5MjJEMEZBQzY4N0M4Q0JGRkJCNkNGQUE1MEIyQUY0MDkzMjFGNThEMEZBQ0Y4QkZGQUQ2ODZEQTY5NEIwNjQwOTQxNkRCNkRCNkRCQURCRkY5RTVBNjVEMkJEQjdGNDA5NTBCQzE0RTVFMEE2MUJGRjhERjYwOEI3NDg2ODA0MDk2MDBBNzJGMDUzOTE1QkZGN0JCRjE5MUFDNUY0ODQwOTZGNThEMEZBQzY3Q0FCRkY2NkQyRDc2NTkyRjE0NDA5N0VBNzJGMDUzOTgzNkJGRjREMkZERkI0QTg2RTE0MDk4REY1OEQwRkFDNkVCQkZGMjcwRjJDQTZCMjAyRDQwOTlENDNFQjFBMUY1OUYzRkZGOTU0MzVBQTI4OTk3NDA5QUM5MjQ5MjQ5MjQ1MzNGRkVFNzJFQTE0NkE5MUM0MDlCQkUwQTcyRjA1MzA4M0ZGRTMyMzI0NzgzNTZCQzQwOUNCMkYwNTM5NzgxQkMzRkZENzU1MkEyRDYzQjQ2NDA5REE3RDYzNDNFQjIyOTNGRkNBRjUzOEEwOEVBNUQ0MDlFOUNCQzE0RTVFMEREM0ZGQkRFOTk5QTM4NjdGMDQwOUY5MUExRjU4RDBGOTEzRkZCMDBGQzdDMDc1MDkzNDBBMDQzNDNFQjFBMUYyMzNGRkExMzc3Q0YyQjBFNUM0MEEwQkRCNkRCNkRCNjdEM0ZGOTExOUFBMzFGNUY3NTQwQTEzODI5Q0JDMTRERDczRkY3RjQ1Nzc3RjNERUI0NDBBMUIyOUNCQzE0RTYwRTNGRjZBRjI3NURFNkRCQjg0MEEyMkQwRkFDNjg3RDY4M0ZGNTI3MTYxQ0I3N0IxNjQwQTJBNzgyOUNCQzE0QzIzRkYzMDMyQ0Y0RUQyMjIyNDBBMzIxRjU4RDBGQUMxQ0JGRkZCNEE2MzEyNTY4QjE0MEEzOUM2ODdENjM0Mzc2QkZGRjA3QzAzRTY1MERDNjQwQTQxNkRCNkRCNkRCQURCRkZFNTQxQTk1RUNDRUMwNDBBNDkxNEU1RTBBNzMwN0JGRkQ5OEM0MTcxRkIyMjA0MEE1MEJDMTRFNUUwQTYxQkZGQ0Q0OEVDQjlGMEYzODQwQTU4NjM0M0VCMUExQkJCRkZDMDVGMzdDRTE4ODFBNDBBNjAwQTcyRjA1MzkxNUJGRkIyQUU4NUU3MUZDRTg0MEE2N0IxQTFGNThEMDcwQkZGQTQwOUE1ODgzRkY0RTQwQTZGNThEMEZBQzY4QTZCRkY5NDJFOUNCQjk1RjVGNDBBNzcwMDAwMDAwMDAwMEJGRjgyQjZCMkRBOERFMEMtSSdDT0xPVVJHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JkkkUkdCRzYiJCIjNSEiIiQiIiEiIiEkIiIhIiIhLUkrQVhFU0xBQkVMU0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkUSJ4NiJRInk2Ig==

By experimenting with other initial conditions, one can see that all nontrivial solutions converge very rapidly to this same limit cycle. Finally, let us check what was computed in the aborted attempt which used the default non-stiff method. We see that it was unable to compute beyond approximately NiMvJSJ4RyIiJw== . plots[odeplot](soln[0], [x, y(x)]);

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