Chapter 17.2, Problem 19E

### 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.19. 4y" + y = cos x

(a)

To determine

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

Explanation

Given data:

The differential equation is,

4yâ€³+y=cosx (1)

Consider the auxiliary equation.

4r2+1=0 (2)

Roots of equation (2) are,

r=âˆ’0Â±(0)2âˆ’4(4)(1)2(4)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€‰}=Â±i48=Â±12i

Write the expression for the complementary solution of two complex roots r=Î±Â±iÎ² ,

yc(x)=eÎ±x(c1cosÎ²x+c2sinÎ²x) (3)

Substitute 0 for Î± and 12 for Î² in equation (3),

yc(x)=e0x(c1cos12x+c2sin12x)

yc(x)=c1cos(12x)+c2sin(12x) (4)

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

yp(x)=Acosx+Bsinx (5)

Differentiate equation (5) with respect to x,

yâ€²p(x)=ddx(Acosx+Bsinx)

yâ€²p(x)=âˆ’Asinx+Bcosx (6)

Differentiate equation (6) with respect to x,

yâ€³p(x)

(b)

To determine

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

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