   Chapter 17.2, Problem 4E

Chapter
Section
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,

y2y+2y=x+ex (1)

Consider the auxiliary equation is,

r22r+2=0 (2)

Roots of equation (2) are,

r=(2)±(2)24(1)(2)2(1){r=b±b24ac2afortheequationofar2+br+c=0}=2±i22=1±i

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

yc(x)=eαx(c1cosβx+c2sinβx)

Substitute 1 for α and 1 for β ,

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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