Chapter 17.2, Problem 4E

### 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 or initial-value problem using the method of undetermined coefficients.4. y" – 2y' + 2y = x + ex

To determine

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

Explanation

Given data:

The differential equation is,

yâ€³âˆ’2yâ€²+2y=x+ex (1)

Consider the auxiliary equation is,

r2âˆ’2r+2=0 (2)

Roots of equation (2) are,

r=âˆ’(âˆ’2)Â±(âˆ’2)2âˆ’4(1)(2)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€‰}=2Â±i22=1Â±i

Write the expression for the complementary solution for the complex roots,

yc(x)=eÎ±x(c1cosÎ²x+c2sinÎ²x)

Substitute 1 for Î± and 1 for Î² ,

yc(x)=e1x(c1cosx+c2sinx)

yc(x)=ex(c1cosx+c2sinx) (3)

Since G(x)=x+ex is a polynomial degree of 1.

If Right hand side (RHS) of a differential equation contains an addition of an exponential function and 1st order polynomial. So, the particular solution yp(x) for this case can be expressed as follows

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