1550078481-Ordinary_Differential_Equations__Roberts_

68 Ordinary Differential Equations

THEOREM SOLUTION OF THE NONHOMOGENEOUS

LINEAR FIRST-ORDER DIFFERENTIAL

EQUATION

If a(x) and b(x) are continuous functions on the interval (a.,(3), then

the solution on (a, (3) of the nonhomogeneo us linear first-order differential

equation

(28) y' = a(x)y + b(x)

is

(36) y(x) = Y1(x)(I< +v(x))

where K is an arbitrary constant, y 1 (x) = er a(t) dt, and v(x) = J x b((t)) dt.

Y1 t

From equation (36), we observe that in order to write expli citly the solu-

tion of the nonhomogeneous linear first-order differential equ ation (28) as

an elementary function, we must be able to write y 1 ( x) = ef x a( t) dt and

v( x) = r b((t )) dt as elementary functions. Of course, being able to write
Y1 t
both of these integrals in terms of elementary functions is often very difficult
or even impossible. Also notice by setting b(x) = 0 in equation (36) reduces
this equation to equation (25), the so lution of the homogeneous linear differ-
ential equation. Thus, equation (36) is an expression for the solution of both
the homogeneous and nonhomogeneous linear first-order differential equation.

EXAMPLE 5 Solution of a Nonhomogeneous Linear IVP

a. Find the so lution of the linear differential equation

(37) y' = (tanx)y + sinx.

b. Solve the initial value problem

(38)

7f (;;

y' = (tanx)y+sinx; Y(4) = v2

and specify the interval on which the solution exists.