   Chapter 3.1, Problem 68E ### 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

# The equation y" + y' – 2y = x2 is called a differential equation because it involves an unknown function y and its derivatives y' and y". Find constants A, B. and C such that the function y = Ax2 + Bx + C satisfies this equation. (Differential equations will he studied in detail in Chapter 9.)

To determine

To find: The function y.

Explanation

Given:

The differential equation is y+y2y=x2 and the function y=Ax2+Bx+C, where A, B, and C are constant.

Derivative rules:

(1) Constant multiple rule: ddx(cf)=cddx(f)

(2) Power rule: ddx(xn)=nxn1

(3) Sum rule: ddx(f+g)=ddx(f)+ddx(g)

Calculation:

Obtain the first derivative of y=Ax2+Bx+C.

y(x)=ddx(Ax2+Bx+C)

Apply the sum rule (3) and the constant multiple rule (1),

Since the derivative of constant function is zero, then ddx(C)=0.

Apply the power rule (2) and simplify the terms,

y(x)=A(2x21)+B(1x11)=A(2x)+B(1)=2Ax+B

Therefore, the first derivative of y is 2Ax+B.

Obtain the second derivative of y=Ax2+Bx+C.

y(x)=ddx(dydx)=ddx(2Ax+B)

Apply the sum rule (3) and the constant multiple rule (1),

Since derivative of constant function is zero, then ddx(C)=0.

Apply the power rule (2) and simplify the terms,

y(x)=2A(1x11)=2A(1)=2A

Therefore, the second derivative of y is 2A

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