APPENDIX

Ordinary

Differential

Equations

PDE models are frequently solved by reducing them to one or more ODEs. This appendix contains a brief review of how to solve some of the basic ODEs that are encountered in this book. At the end of the appendix are several exercises that should be solved by hand; the reader might want to check the solutions using a computer algebra package.

For notation, we let y = y( x) be the unknown function. Derivatives will be denoted by primes, i.e., y' = y'(x), y" = y"(x). Sometimes we use the differential notation y' = ~. Iff is a function, an antiderivative is defined as a function F whose derivative is f, i.e., F'(x) = f(x). Antiderivatives are unique only up to an additive constant, and they are often denoted by the usual indefinite integral sign:

F(x) = f f(x)dx.

Usually an artibrary constant of integration C is added to the right side. However, in this last expression, it is sometimes impossible to evaluate the antiderivative F at a particular value of x. For example, if f(x) = sin xl x, then there is no simple formula for the antiderivative; that is,

f sinx F(x) = -x-dx

cannot be expressed in closed form in terms of elementary functions, and thus we could not find, for example, F(2). Therefore, it is usually better to denote the antiderivative by an integral with a variable upper limit,

F(x) = 1x f(s)ds,

169

170 A. Ordinary Differential Equations

where a is any constant. By one form of the fundamental theorem of calculus, F'(x) = tex). Now, for example, the antiderivative of sin x/x can be written

and easily we find that

1x sins F(x) = -ds,

o s

12 sins F(2) = -ds.

o s

First-Order Equations An ODE of the first order is an equation ofthe form

G(x,y,y') = o. There are three types of these equations that occur regularly in PDEs: separable, linear, and Bernoulli.

Separable Equations

A first-order equation is separable if it can be written in the form

dy dx = tex)g(y).

In this case we separate variables to write

dy g(y) = f(x)dx.

Then we can integrate both sides to get

j dy = jf(X)dx + C, g(y)

which defines the solution implicitly. As noted above, sometimes the anti­derivatives should be written as definite integrals with a variable upper limit of integration.

The simplest separable equation is the growth-decay equation

y' = 'Ay,

which has general solution

y = ceAx.

The solution models exponential growth if 'A > 0 and exponential decay if A < o.

A. Ordinary Differential Equations 171

Linear Equations

A first-order linear equation is one of the form

y' + p(X)y = q(x).

This can be solved by multiplying through by an integrating factor of the form

This turns the left side of the equation into a total derivative, and it becomes

! (y exp(l x P(S)ds)) = q(x) exp(lx

p(s)ds)

Now, both sides can be integrated from a to x to find y. We illustrate this procedure with an example. EXAMPLE

Find an expression for the solution to the initial value problem

y' + 2xy =../x, yeO) = 3.

The integrating factor is expC!; 2sds) = exp(x2). Multiplying both sides of the equation by the integrating factor gives

..x2 , I.. x2 (yt; ) = yxe .

Now, integrating from 0 to x (while changing the dummy variable of integration to s) gives

Solving for y gives

2 2 2 lox = 3e-x + 0 ..jSes -x ds.

As is frequently the case, the integrals in this example cannot be com­puted easily, if at all, and we must write the solution in terms of integrals with variable limits. 0

Bernoulli Equations

Bernoulli equations are nonlinear equations having the form

y' + p(x)y = q(x)yn.

172 A. Ordinary Differential Equations

The transformation of dependent variables w = yl-n turns a Bernoulli equation into a linear equation for w.

Second-Order Equations Special Equations

Some second-Qfder equati9ns cap. be i111megiately reduced to a first order equation. FQr exarnple, if the equation has the mrIn

G(x, y', y") :::; 0,

where y is missing, then the substitution v :;:; y' reduces the equation to the first-order equation

G(x, v, v') :;:: o. If the second-order eq~ati~n does not d~pe~q explicitly OIl; t:p.e ~ndepe1'\­dent variable x, that is, it has the form

G(y, y', y") = 0,

then we again define v = y'. Then

11 d, dv y = -y = ax ax

So the equation becomes

dv G(y, v, dy v) :::; 0,

which is a first-order equation in v = v(y).

d;v -v. dy

Linear, Constant-Coefficient Equations

The equation

ay" + by' + cy = 0,

where a, b, and c are constants, occurs frequently in applications. If we try a solution of the form y = ernx , where m is to be determined, then substitution into the equation gives the so-called characteristic equation

amz + bm + c = 0

for m. This is a quadratic polynomial that will have two roots, ml and mz. Three possibilities can occur: unequal real roots, equal real roots, and complex roots (which must be complex conjugates).

Case (I). ml, mz real and unequal. In this case two independent solutions are em,x and emzx .

A, Ordinary Differential Equations 173

Case (II). ml, m2 real and equal, i.e., ml = m2 :; m. In this case two independent solutions are ernx and xernx •

Case (III). ml = a + ifJ, m2 = a - ifJ are complex conjugate roots. In this case two real, independ~nt solutions are fPlC sin fJx and fPlC cos fJx.

We recall that the general solution of a linear equation is a linear combination of two independent solutions.

Of particular importance are the two equations y" + a2y = 0, which has general solution y = CI cos ax + C2 sin ax, and y" - a2y = 0, which has general solutioll y = CI e-ax + C2~. These two equations occur so frequently thClt it is best to memorize them.

Cauchy-Euler Equations

It is difficult to solve second-order linear equations with variable coeffi­cients. Often, the reader may recall, power series methods are applied. However, there is a special equation that can be solved with simple formulae, namely, a Cauchy-Euler equation of the form

ax2y" + bxy' + cy = o. This equation admits power functions as solutions. Hence, if we try a solution of the formy = xm , where m is to be determined, then we obtain upon substitution the characteristic equation

am(m-l)+bm+c=O.

This quadratic equation has two roots, ml and m2. Thus, there are three cases:

Case (I). mb m2 real and unequal. In this case two independent solutions are xm1 and xm2.

Case (II). ml, m2 real and equal, i.e., ml = m2 == m. In this case two independent solutions are xm and xm In x.

Case (III). ml = a+ifJ, m2 = a-ifJ are complex conjugate roots. In this case two real, independent solutions are xa sin(fJ In x) and xrx cos(fJ In x).

Particular Solutions The general sQh;J,tion of the inhomogeneous ODE

y" + p(x)y' + q(x)y = [(x)

is

Y = CIYI(X) + C2YZ(X) + yp(x),

where YI and Y2 are independent solutions of the homogeneous equations (when fC x) == 0), and yp is any particular solution to the inhomogeneous equation. For constant-coefficient equations a particular solution can

174 A. Ordinary Differential Equations

sometimes be "guessed" from the form of f(x); the reader may recall that this guessing method is called the method of undetermined coefficients. In any case, however, there is a general formula, called the variation of pa­rameters formula, which gives the particular solution in terms of the two linearly independent solutions Yl and yz. The formula, which is derived in elementary texts, is given by

There are several introductory texts on differential equations (see, for example, Boyce and DiPrima (1995) or Braun (1983)). Birkhoff and Rota (1978) and Waltman (1987) are two more advanced texts.

Exercises Solve the following differential equations.

1. y' + 2y = e-x .

2. y' = -3y.

3. y" + Sy = o.

5. x2y" - 3xy' + 4y = O.

6. y" + xy,2 = O.

7. y" + y' + Y = O.

S. yy" - y,3 = O.

9. 2x2y" + 3xy' - Y = O.

10. y" - 3y' - 4y = 2 sin x.

11. y" + 4y = x sin 2x.

12. y' - 2xy = l.

13. y" + 5y' + 6y = O .

. )

1 ' x-4. Y = 1+3y3'

Exercises

TABLE OF LAPLACE TRANSFORMS

u(t)

sin at

cos at

sinh at

cosh at

H(t - a)u(t - a)

1 - erf (~) At

e~a21(4t)

J4t3

(u * v)(t)

tu(t)

u(t)/t

u(at)

o(t - a)

U(s)

-L s-a

s a2+s2

s s2_a2

U(s - a)

U(s)V(s)

-U'(s)

Isoo U(r)dr

U(s/a)/a

175

References

1. M. Abramowitz and LA. Stegun, eds., Handbook of Mathematical Functions, Dover Publications, New York (1965).

2. R.B. Bird, W.E. Stewart, and E.N. Lightfoot, 7Yansport Phenomena, John Wiley and Sons, New York (1960).

3. G. Birkhoff and J.C. Rota, Ordinary Differential Equations, John Wiley and Sons, New York (1978).

4. W.E. Boyce and R.C. DiPrima, Elementary Differential Equations, 5th ed., John Wiley and Sons, New York (1995).

5. M. Braun, Differential Equations and Their Applications, 3rd ed., Springer-Verlag, New York (1983).

6. H.S. Carslaw and J.C. Jaeger, Conduction of Heat in Solids, 2nd ed., Clarendon Press, Oxford (1959).

7. A.J. Chorin and J.E. Marsden, A Mathematical Introduction to F1uid Mechanics, 3rd ed., Springer-Verlag, New York (1993).

8. R.v. Churchill, Fourier Series and Boundary Value Problems, 2nd ed., McGraw-Hill, New York (1958).

9. R.V. Churchill, Complex Variables and Applications, 2nd ed., McGraw-Hill, New York (1960).

10. R.V. Churchill, Operational Mathematics, 3rd ed., McGraw-Hill, New York (1972).

11. G. Folland, Introduction to Partial Differential Equations, 2nd ed., Princeton Univ. Press (1996).

12. P. Grindrod, Patterns and Waves, Clarendon Press, Oxford (1991).

13. C.w. Groetsch, Inverse Problems in the Mathematical Sciences, Vieweg, Braun -schweig/Wiesbaden (1993).

14. R.B. Guenther and J.W. Lee, Partial Differential Equations of Mathematical Physics and Integral Equations, Dover Publications, New York (1992).

15. F. John, Partial Differential Equations, 4th ed., Springer-Verlag, New York (1982).

177

178 References

16. C.C. Lin and L.A. Segel, Mathematics Applied to Deterministic Models in the Natural Sciences, SIAM, Philadelphia (1989).

17. J.D. Logan, Introduction to Nonlinear Partial Differential Equations, Wiley-Interscience, New York (1994).

18. J.D. Logan, Applied Mathematics: Second edition, Wiley-Interscience, New York (1997).

19. G. de Marsily, Quantitative Hydrogeology, Academic Press, New York (1987).

20. R. McOwen, Partial Differential Equations, Prentice-Hall, Englewood Cliffs NJ (1995).

21. K.w. Morton and D.F. Mayers, Numerical Solution to Partial Differential Equations, Cambridge Univ. Press (1994).

22. J.D. Murray, Mathematical Biology, Springer-Verlag, New York (1987).

23. M. Renardy and R.C. Rogers, An Introduction to Partial Differential Equations, Springer­Verlag (1993).

24. J. Robertson, Engineering Mathematics with Maple, McGraw-Hill, New York (1996).

25. L.A. Segel, Mathematics Applied to Continuum Mechanics, Dover Publications, New York (1987).

26. J. Smoller, Shock Waves and Reaction-Diffusion Equations, 2nd ed., Springer-Verlag (1995).

27. W. Strauss, Introduction to Partial Differential Equations, John Wiley and Sons, New York (1992).

28. R. Strichartz, A Guide to Distribution Theory and Fourier 1Yansforms, CRC Press, Boca Raton (1994).

29. A.N. 1Jchonov and A.A. Samarskii, Equations of Mathematical PhYSics, Dover Publica­tions, New York (1990).

30. P. Waltman, A Second Course in Elementary Differential Equations, Academic Press, New York (1987).

31. G.B. Whitham, Linear and Nonlinear Waves, Wiley-Interscience, New York (1974).

Index

acoustics, 42 acoustic approximation, 43 adsorption, 21 Airy function, 84, 115 antiderivative, 169 applications to

acoustics, 42-44 age-structured populations,

90 bacterial growth, 19 chemical reactors, 156 contaminant transport in

aquifers, 20, 74, 131 determining thermal

conductivity, 154 electrostatics, 4, 41 equilibrium temperatures, 34, 41 guitar strings, 24-28, 155 heat transfer, 3, 16, 18, 31,

50-55, 63, 77 nerve impulses, 156 nutrient adsorption, 14 population growth, 2, 155 quantum mechanics, 28-31, 83,

102 signal processing, 105-107 temperature of earth, 68, 143

temperatures in disks and spheres, 40, 139-143

traffic flow, 115 wave propagation, 56, 60, 65

aquifer, 20 atom, 157

Bernoulli equation, 171 Bessel's inequality, 99 Bessel's differential equation, 145 Bessel functions, 145 boundary condition, 3-4 boundary value problem, 4 Burgers' equation, 24

canonical form, 47-48 Cauchy problem, 50, 56 Cauchy-Euler equation, 173 Cauchy-Schwartz inequality, 102 characteristics, 12, 58 characteristic coordinates, 13-14,

47 characteristic parallelogram, 60 complete, 100 co"nservation law, 9-10, 32 constitutive relation, 11 convection, 11

179

180

convection equation, 11 convection-diffusion, 17 convergence in L2, 97 convergence in mean square, 97 convergence, pointwise, 97 convergence, uniform, 97 convolution, 76, 80 Courant-Friedrichs-Levy (CFL)

condition, 162

d'Alembert's formula, 57 decay, 12 diffusion, 15 diffusion constant, 16 diffusion equation, 16, 33 diffusivity, 17 dimensionless variable, 51 Dirichlet condition, 8, 35, 124 Dirichlet problem, 4 Dirichlet principle, 36, l37, 138 discriminant, 45 dispersion, 183 dispersion constant, 21 dispersion equation, 22 divergence operator, 33 divergence theorem, 33, l35 domain of dependence, 58 Duhamel's principle, 69, 149

eigenfunction, 110 eigenvalue, 110 elliptic, 45, 47 energy argument, 113 energy integral, 28, 137 energy spectrum, 100, 104 error, mean square, 97 error, pointwise, 97 Euler's method, 158

Fick's law, 16, 33 finite difference method, 158 flux, 9, 32 Fourier coefficients, 98, 103 Fourier heat law, 16,33 Fourier integral theorem, 79 Fourier series, 94, 103 Fourier series, generalized, 98

Fourier transform, 78 frequency spectrum, 104 Freundlich isotherm, 22 fundamental solution, 54

Index

Gauss-Seidel method, 165 Gram-Schmidt orthogonalization,

101 Green's function, 55 Green's identities, 35, 136

Haar wavelets, 102 heat equation, 3, 17 heat kernel, 54 Hermite polynomials, 101 hyperbolic equation, 45-46

initial boundary value problem, 4 initial condition, 4

inverse problem, 152

Korteweg-de Vries equation, linearized, 83

L2[a, bj, 95 Langmuir isotherm, 22 Laplace transform, 72 Laplace transforms, table of, 175 Laplace's equation, 34, 36 Laplacian, 34 Laplacian in other coordinate

systems, 39 Legendre polynomials, 101, 143 Legendre's differential equation,

143 Leibniz's rule, 8 light cone, 58 linearity, 4 Liouville transformation, 30

Malthus' law, 2 maximum principle, 38 Maxwell's equations, 41

Neumann condition, 18, 35, 124 nonlinear, 4 norm, 95

Index

order of an equation, 4 orthogonal, 94-95 orthonormal, 96

parabolic equation, 45, 47 Parseval's equality, 100, 104 periodic boundary condition, 131 periodic function, 105 pi theorem, 51 piecewise continuous, 106 piecewise smooth, 106 Poisson's equation, 34 Poisson's integral formula, 135 Poisson's integral representation,

55 porosity, 20 principle part of an equation, 45

quantum mechanics, 28

radiation condition, 18, 124 reaction-convection equation, 14 region of influence, 58 retardation constant, 24 Robin condition, 18, 124

saturated medium, 20

scalar product, 95 Schrodinger equation, 29-30 Schwartz class, 78

181

separable equation, 170 separation constant, 117 separation of variables, 108, 116 solution to a PDE, 5 sound speed, 43 square-integrable, 95 stability, 62, 162 standing wave, 28 Sturm-Liouville problem, 110 support of a function, 57

traveling wave, 11, 22

variation of parameters formula (ODE), lSI, 174

variation of parameters formula (PDE),70

vibrations of a string, 24

wave-front, 22 wave equation, 27 wave function, 29 wave speed, 26 well-posed, 62

Undergraduate Texts in Mathematics

(continued from page ii)

James: Topological and Uniform Spaces.

Janich: Linear Algebra. Janich: Topology. Kemeny/Snell: Finite Markov Chains. Kinsey: Topology of Surfaces. Klambauer: Aspects of Calculus. Lang: A First Course in Calculus. Fifth

edition. Lang: Calculus of Several Variables.

Third edition. Lang: Introduction to Linear Algebra.

Second edition. Lang: Linear Algebra. Third edition. Lang: Undergraduate Algebra. Second

edition. Lang: Undergraduate Analysis. LaxIBursteinlLax: Calculus with

Applications and Computing. Volume 1.

LeCuyer: College Mathematics with APL.

LidllPilz: Applied Abstract Algebra. Second edition.

Logan: Applied Partial Differential Equations.

Macki-Strauss: Introduction to Optimal Control Theory.

MaUtz: Introduction to Mathematical Logic.

Marsden/Weinstein: Calculus I, II, III. Second edition.

Martin: The Foundations of Geometry and the Non-Euclidean Plane.

Martin: Geometric Constructions. Martin: Transformation Geometry: An

Introduction to Symmetry. Millman/Parker: Geometry: A Metric

Approach with Models. Second edition.

Moschovakis: Notes on Set Theory. Owen: A First Course in the

Mathematical Foundations of Thermodynamics.

Palka: An Introduction to Complex Function Theory.

Pedrick: A First Course in Analysis.

PeressinilSullivanlUhl: The Mathematics of Nonlinear Programming.

PrenowitzlJantosciak: Join Geometries. Priestley: Calculus: A Liberal Art.

Second edition. ProtterlMorrey: A First Course in Real

Analysis. Second edition. Protter/Morrey: Intermediate Calculus.

Second edition. Roman: An Introduction to Coding and

Information Theory. Ross: Elementary Analysis: The Theory

of Calculus. Samuel: Projective Geometry.

Readings in Mathematics. Scharlau/Opolka: From Fermat to

Minkowski. Sethuraman: Rings, Fields, and Vector

Spaces: An Approach to Geometric Constructability.

Sigler: Algebra. Silverman/Tate: Rational Points on

Elliptic Curves. Simmonds: A Brief on Tensor Analysis.

Second edition. Singer: Geometry: Plane and Fancy. Singer/Thorpe: Lecture Notes on

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Smith: Linear Algebra. Third edition. Smith: Primer of Modem Analysis.

Second edition. Stanton/White: Constructive

Combinatorics. Stillwell: Elements of Algebra:

Geometry, Numbers, Equations. Stillwell: Mathematics and Its History. Stillwell: Numbers and Geometry.

Readings in Mathematics. Strayer: Linear Programming and Its

Applications. Thorpe: Elementary Topics in

Differential Geometry. Toth: Glimpses of Algebra and

Geometry. Readings in Mathematics.

Undergraduate Texts in Mathematics

Troutman: Variational Calculus and Optimal Control. Second edition.

Valenza: Linear Algebra: An Introduction to Abstract Mathematics.

WbyburnJDuda: Dynamic Topology. Wilson: Much Ado About Calculus.