Chapter 17.2, Problem 10E

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8th Edition

James Stewart

ISBN: 9781285741550

Textbook Problem

Solve the differential equation or initial-value problem using the method of undetermined coefficients.

**10.** *y"* + *y'* – 2*y* = *x* + sin 2*x, y*(0) = 1, *y'*(0) = 0

To determine

**To solve:** The differential equation or initial-value problem by the method of undetermined coefficients.

Explanation

**Given data:**

The differential equation is,

with

Consider the auxiliary equation is,

Roots of equation (2) are,

Write the expression for the complementary solution,

Substitute 1 for

The particular solution

Differentiate equation (3) with respect to *x*,

Differentiate equation (4) with respect to *x*,

Re-write equation (1) by neglecting the term

Substitute equations (3), (4) and (5) in equation (6),

Differentiate the equation (7) with respect to *x*,

Simplify the equation,

Substitute 0 for *x* in equation (7),

Substitute
*A* in equation (8),

Substitute
*A* and
*B* in equation (3),

Re-write equation (1) by neglecting *x*.

The particular solution

Differentiate equation (11) with respect to *x*,

Differentiate equation (12) with respect to *x*,

Substitute equations (11), (12), and (13) in equation (10),

Substitute 0 for *x* in equation (14)

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