   Chapter 17.2, Problem 21E

Chapter
Section
Textbook Problem

Solve the differential equation using (a) undetermined coefficients and (b) variation of parameters.21. y" – 2y' + y = e2x

(a)

To determine

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

Explanation

Given data:

The differential equation is,

y2y+y=e2x (1)

Consider the auxiliary equation.

r22r+1=0 (2)

Roots of equation (2) are,

r=(2)±(2)24(1)(1)2(1){r=b±b24ac2afortheequationofar2+br+c=0}=22=1

Write the expression for the complementary solution of the one real root.

yc(x)=c1erx+c2xerx (3)

Substitute 1 for r in equation (3),

yc(x)=c1e1x+c2xe1x

yc(x)=c1ex+c2xex (4)

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

yp(x)=Ae2x (5)

Differentiate equation (5) with respect to x.

yp(x)=A(2)e2x

yp(x)=2Ae2x (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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