   Chapter 3.3, Problem 53E ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

#### Solutions

Chapter
Section ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

# Find constants A and B such that the function y = A sin x + B cos x satisfies the differential equationy" + y' – 2y = sin x.

To determine

To find: The constants A and B.

Explanation

Given:

The function is y=Asinx+Bcosx.

The differential equation y+y2y=sinx.

Derivative Rules:

(1) Sum Rule: ddx[f(x)+g(x)]=ddx[f(x)]+ddx[g(x)]

(2) Difference Rule: ddx[f(x)g(x)]=ddx[f(x)]ddx[g(x)]

Calculation:

Obtain the first derivative of y.

y=ddx(y)=ddx(Asinx+Bcosx)

Apply the sum rule (1),

Thus, the first derivative of y is y=AcosxBsinx.

Obtain the second derivative of y.

y=ddx(y)=ddx(AcosxBsinx)

Apply the difference rule (2),

Thus, the first derivative of y is y=AsinxBcosx

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