   Chapter 17, Problem 14RE

Chapter
Section
Textbook Problem

Solve the initial-value problem.14. 9y" + y =3x + e-x, y(0) = 1, y'(0) = 2

To determine

To solve: The initial-value problem.

Explanation

Given data:

The initial-value problem is,

9y+y=3x+ex (1)

With y(0)=1 and y(0)=2 .

Consider the auxiliary equation.

9r2+1=0 (2)

Roots of equation (2) are,

r=0±(0)24(9)(1)2(9){r=b±b24ac2afortheequationofar2+br+c=0}=±i618=±13i

Write the expression for the complementary solution of two complex roots r=α±iβ ,

yc(x)=eαx(c1cosβx+c2sinβx) (3)

Substitute 0 for α and 13 for β in equation (3),

yc(x)=e0x(c1cos13x+c2sin13x)

yc(x)=c1cos13x+c2sin13x (4)

Re-arrange equation (1) by neglecting the term ex ,

9y+y=3x (5)

The Right hand side (RHS) of a differential equation contains a polynomial. Therefore, the trail solution yp(x) for this case can be expressed as follows.

yp1(x)=Ax+B (6)

Differentiate equation (6) with respect to x,

yp1(x)=ddx(Ax+B)

yp1(x)=A (7)

Differentiate equation (7) with respect to x,

yp2(x)=ddx(A)

yp1(x)=0 (8)

Substitute equations (6) and (8) in equation (1),

9(0)+Ax+B=3x

Ax+B=3x (9)

By equating coefficients of equation (9), consider the followings,

A=3

And

B=0

Substitute 3 for A and 0 for B in equation (6),

yp1(x)=3x+0

yp1(x)=3x (10)

Re-arrange equation (1) by neglecting the term 3x ,

9y+y=ex (11)

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

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