APPENDIX Ordinary Differential Equations PDE models are frequ ently solved by reducing th em to on e 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 func tion . Deri vativ es will be denoted by primes, i.e., y' = y'(X), y" = y"(X). Sometimes we use the differ ent ial notation y' = '!t .If f is a function, an antideriuatiue is defined as a function F whose derivative is i, i.e., F'(x) = fex) . Antiderivatives are unique only up to an additive constant, and they are often denot ed by the usual indefinite integral sign: F(x) = f j(x )dx + C. An arbitrary constant of int egration C is added to the right side. How- ever, in this last expression, it is someti mes impossible to evaluate the antid erivative F at a particular value of x. For example, if [(x) = sin x /x , then th ere is no simple formula for the antiderivative; that is, f si nx F(x ) = -- dx x cannot be expressed in closed form in te rms ofelementary fun ctions, and thus we could not find , for example, F(Z). Th erefore, it is b etter to denot e the antiderivative by an integral with a variable upper limit, F(x ) = f' f(s )ds + C, "
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APPENDIX
OrdinaryDifferentialEquations
PDE models are frequ ently solved by reducing them to one or more ODEs.This appendix contains a bri ef review of how to solve som e of the basicODEs that are encountered in this book. At the end of th e appendix areseveral exercises that should be solved by hand; the reader might want tocheck the solutions using a compute r algebra package .
For notation, we let y = y (x) be the unknown func tion . Derivatives willbe denoted by pr imes , i.e ., y ' = y'(X), y" = y"(X). Sometimes we use thedifferential notation y' = '!t .If f is a function, an antideriuatiue is definedas a fun ction F whose derivative is i, i.e., F'(x) = fex) . Antiderivativesare unique onl y up to an additive constant, and they are often denotedby the usual indefinite integral sign:
F(x) = f j(x )dx + C.
An arbitrary constant of integration C is added to the right side. Howeve r, in this last expression, it is sometimes impossible to evaluate th eantiderivat ive F at a particular value of x . For example, if [(x) = sin x /x ,then there is no simple formula for the antiderivative; that is,
f sinxF(x ) = -- dx
x
cannot be expressed in closed form in terms of eleme ntary fun ctions, andthus we could not find , for example, F(Z). Th erefore, it is better to denoteth e antiderivative by an integral with a variable upp er limit,
F(x ) = f'f(s )ds + C,
"
198 A. Ordinary Differential Equations
where a is any constant (observe that a and C are not independent, sincechanging one changes the other) . By the fundamental theorem of calculus , F '(x) = f(x)' Now, for example , the ant iderivative of sin x /x can bewritten (taking C = 0)
1x sin s
F(x) = -ds,o s
and easily we find that
12 sin sF(Z) = - ds ~ 1.605.
o s
First-Order EquationsAn ODE of the first order is an equation of the form
G(x , y, y/) = O.
There are three types of these equations that occur regularly in PDEs:separable, linear, and Bernoulli . The general solution involves an arbitrary constant C that can be determined by an initial condition of theform y(xo) = Yo .
Separable Equations
A first-order equation is separable if it can be written in the form
dydx = f(x)g(y) .
In this case we separate variables to write
dy = f(x)dx .g(y )
Then we can integrate both sides to get
! dy = !f(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 upperlimit of integration.
The simplest separable equation is the growth-decay equation
y/ = )"y ,
which has general solution
y = ce":
A . Ordinary Differential Equations 199
The solution mod els exponentia l growth if A > 0 and expone ntial decayin < o.
Linear Equations
A first-order linear equa tion is one of the form
.11' + p(x )y = q(x ).
This can be solved by multiplying through by an int egrating factor of theform
e.Cp (s)ds .
This turns the left side of the equa tion into a total derivative, and itbecomes
~ ( .11 expel ' P(S)dS)) = q(x) exp(l x
p(s)ds)
Now, both sides can be integrated from (l to x to find y . We illustra te th isprocedure with an example.E XAMP LE
Find an expression for the solutio n to the initial value problem
.11' + 2xy = JX, .11( 0) = 3.
The integrating factor is exp(j~Y 2sds) = exp(x2) . Multip lying bo th sidesof the equat ion by the integrati ng factor gives
(ye")' = JXe x: .
Now, integrating from () to X (while changing the dummy var iable ofintegration to s) gives
Solving for .11 gives
y(x) = e- X'(3 + r JSe s'ds) = 3e- x' + r JSes 2-
x' ds.10 10As is freque ntly the case, the integrals in this example cannot be computed eas ily, if at all, and we mu st write the solution in term s of integralswith variable limi ts. D
Bernoulli Equations
Bernoulli equations are nonlinear equations having the form
.11' + p(x)y = q(x)yn.
200 A. Ordinary Differential Equations
The transformation of dependent variables w = yl-n turns a Bernoulliequation into a first-order linear equation for w.
Second-Order EquationsSpecial Equations
Some second-order equations can be immediately reduced to a firstorder equation. For example, if the equation has the form
G(x, y', y") = 0,
where y is missing, then the substitution v = y' reduces the equation tothe first-order equation
G(x , v , v') = 0.
If the second-order equation does not depend explicitly on the independent variable x, that is, it has the form
G(y, y', y") = 0,
then we again define v = y' . Then
" d,y =-ydx
So the equation becomes
dvdx
dv dy
dy dxdvdy v.
dvG(y , v , dy v) = 0,
which is a first-order equation in v = v(y).
Linear, Constant-Coefficient Equations
The equation
ay" + by' + cy = 0,
where a, b, and c are constants, occurs frequently in applications. Werecall that the general solution of a linear, second-order, homogeneousequation is a linear combination of two independent solutions. That is, ifYl(X) and yz(x) are independent solutions, then the general solution is
y = ClYl(X) + czYz(x),
where Cl and Cz are arbitrary constants. If we try a solution of the formy = emx , where m is to be determined, then substitution into the equationgives the so-called characteristic equation
am' + bm + c = °
A. Ordinary Differential Equations 201
for m . This is a quadratic polynomial that will have two roots, m j and mz.Three poss ibiliti es can occur: un equal real roots , equal real roots, andcomplex roots (whi ch mu st be complex conjugates).
Case (I). m j, mz rea l and unequal. In this case two independentsolutions are el111X and e111 2X
•
Case (II) . m j , mz real and equal, i.e., m ] = mz == m . In this case twoind ependent solutions are el11X and xel11X
•
Case (III ) . m j = a + i f3, mz = a - if3 are complex conjugate roots. Inthis case two real, ind ependent solutions are e" sin f3x and e" cos f3x .
Of particular imp ortance are the two equations y " + aZy = 0, wh ichhas gene ral soluti on y = C l cos ax + C2 sin ax, and y" - aZy = 0,which has general solution y = C] e- ax + czeax , or equivalently, y =C1 cosh ax + Cz sinh ax . These two equations occur so freque ntly that it isbest to memorize their solutions.
Cauchy-Euler Equations
It is difficult to solve second-order linear equations with variable coefficients. Often , the reader may recall, power seri es methods are applied .Howeve r, there is a special equation that can be solved with simpleformulae, na mely , a Cauchy-Euler equation of the form
ax2y" + bxy' + cy = O.
Th is equation adm its power functions as solutions . Hen ce, if we try asolution of the form y = x'" , where m is to be determined , then we obtainupon substitution the characteristic equation
am(m - 1) + bm + c = O.
This quadratic equation has two roots, m j and mz. Thu s, there are threecases:
Case (I). m ], mz real and unequal. In th is case two independe ntsolutions are X1111 and X 1111
•
Case (II) . m j , mz real and equa l, i.e. , m ] = mz == m. In th is case twoindepende nt solutions are x l11 and x l11 In x .
Case (III ). m j = a + i{3, m2 = a - i{3 are complex conjugate roots . In th iscase two real , independ ent solut ions are XU sin( f3 ln x) and xa cos(f3 ln x) .
Particular SolutionsTh e gene ral solution of the inhomogeneous ODE
y" + p( x )y ' + q(x )y = [(x)
is
202 A. Ordinary Differential Equ ations
where Yl and yz are independent solutions of the homogeneous equat ion[when [(x) == 0], and yp is any particular solution to the inhomogeneousequation. For constant-coefficient equations a particular solutio n cansometimes be "guessed" from the form of [(x); the reader may recall thatthis guessing method is called the method of undetermined coefficients.In any case, however, there is a gen eral formula , called the variation o[parameters formu la , which gives the particular solutio n in terms of the twolinearly independent solutions Yl and yz. The formula, which is derivedin elementary texts, is given by
There are several int rodu ctory texts on differential equations [see,for example, Boyce and DiPrim a (1995) or Edwards and Penney (2004)].Birkhoff and Rota (1978) and Hirsch , Smale, and Devaney (2004) are twomore advanced texts.
ExercisesSolve the following differential equations.
1. y' + 2y = e- x .
2. y' = - 3y .
3 . y" + By = O.
4. y' - xy = XZy z .
5. x2y" - 3xy' + 4y = O.
6 . y" + xy'Z = o.
7. y" + y' +Y = O.
8. yy" - y, 3 = O.
9. 2x2y" + 3xy ' - Y = O.
10. y" - 3y' - 4y = 2 sinx.
11 . y" + 4y = x sin 2x .
12. y' - 2xy = 1.
13. y" + 5y' + 6y = O.
, x2
14. Y = 1+3y3 '
15. y" - 6y = O.
Exercises
TABLE OF LAPLACE TRANSFORMS
u(t)
t"
sin at
cos at
sinh at
cosh at
H(t - a)u(t - a)
1 - erf ( -..E-)J4t
U( s)
---Ls-a
sa'+s'
sS2_a2
U(s - a)
20 3
(u * v)(t)
tu(t)
u(t) / t
u(at)
o(t - a)
U(s )V(s)
- U' (s)
!,OO U(r)dr
U(s / a) / a
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206 References
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19. J .D. Logan , Introdu ction to Nonlinear Partial Differentia l Equations, Wiley-Int erscien ce,New York (1994).
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21. J .D. Logan , 'Transport Modeling in Hydrogeochemical Systems, Springer-Ver lag, New York(2001).
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23. K.W. Morton and D.F. Mayers, Numerical Solution to Partial Differential Equations,Cambridge Univ. Press (1994).
24. J.D . Murray, Math emati cal Biology, Vol. 11, Spring er-Verlag, New York (2003).
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26. L.A Segel, Math ematics Applied to Continuum Mechanics, Dover Publications, New York(1987) .
27. J. Smoller, Shock Waves and Reaction-DiffUsion Equations, 2nd ed., Spring er-Verlag(1995) .
28. W. Strauss, Introduction to Partial Differential Equations, John Wiley and Sons, New York(1992) .
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