Chapter 17.2, Problem 22E

### Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

Chapter
Section

### Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
Textbook Problem

# Solve the differential equation using (a) undetermined coefficients and (b) variation of parameters.22. y" – y' = ex

(a)

To determine

To solve: The differential equation by using method of undetermined coefficients.

Explanation

Given data:

The differential equation is,

yâ€³âˆ’yâ€²=ex (1)

Consider the auxiliary equation.

r2âˆ’r=0 (2)

Roots of equation (2) are,

r=âˆ’(âˆ’1)Â±(âˆ’1)2âˆ’4(1)(0)2(1)â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰â€‰{âˆµr=âˆ’bÂ±b2âˆ’4ac2aforâ€‰theâ€‰equationâ€‰ofar2+br+c=0â€‰â€‰}=1Â±12=0â€‰andâ€‰1

Write the expression for the complementary solution of two real roots.

yc(x)=c1er1x+c2er2x (3)

Substitute 0 for r1 and 1 for r2 in equation (3),

yc(x)=c1e0x+c2e1x

yc(x)=c1+c2ex (4)

If Right hand side (RHS) of a differential equation contains only an exponential function, therefore, the trail solution of the differential equation is also contains exponential function. So, the trail solution yp(x) for this case can be expressed as follows.

yp(x)=Axex (5)

Differentiate equation (5) with respect to x.

yâ€²p(x)=Axex=A(xex+ex(1))

yâ€²p(x)=Aex(1+x) (6)

Differentiate equation (6) with respect to x

(b)

To determine

To solve: The differential equation by using method of variation of parameters.

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