Chapter 17.2, Problem 18E

Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

Chapter
Section

Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
Textbook Problem

Write a trial solution for the method of undetermined coefficients. Do not determine the coefficients.18. y" + 4y = e3x + x sin 2x

To determine

To write: The trail solution for the method of undetermined coefficients.

Explanation

Given data:

The differential equation is,

yâ€³+4y=e3x+xsin2x (1)

Consider the auxiliary equation is,

r2+4=0 (2)

Roots of equation (2) are,

r=âˆ’0Â±(0)2âˆ’4(1)(4)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€‰}=Â±4i2=Â±2i

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

yc(x)=eÎ±x(c1cosÎ²x+c2sinÎ²x)

Substitute 0 for Î± and 2 for Î² ,

yc(x)=e0x(c1cos2x+c2sin2x)=c1cos2x+c2sin2x

Re-write equation (1) by neglecting the term xsin2x ,

yâ€³+4y=e3x (3)

If Right hand side (RHS) of a differential equation contains only exponential function.

Therefore, the trail solution yp2(x) for this case can be expressed as follows.

yp1(x)=Ae3x (4)

Re-write equation (1) by neglecting the term e3x ,

yâ€³+4y=xsin2x (5)

If Right hand side (RHS) of a differential equation contains a product a sinusoidal function, and nth order polynomial, the trail solution of the differential equation can be expressed as follows

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