Chapter 17, Problem 14RE

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8th Edition

James Stewart

ISBN: 9781285741550

Textbook Problem

Solve the initial-value problem.

**14.** 9y*"* + *y* =3*x* + *e*^{-x}, *y*(0) = 1, *y'*(0) = 2

To determine

**To solve:** The initial-value problem.

Explanation

**Given data:**

The initial-value problem is,

With

Consider the auxiliary equation.

Roots of equation (2) are,

Write the expression for the complementary solution of two complex roots

Substitute 0 for

Re-arrange equation (1) by neglecting the term

The Right hand side (RHS) of a differential equation contains a polynomial. Therefore, the trail solution

Differentiate equation (6) with respect to *x*,

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

Substitute equations (6) and (8) in equation (1),

By equating coefficients of equation (9), consider the followings,

And

Substitute
*A* and
*B* in equation (6),

Re-arrange equation (1) by neglecting the term

The Right hand side (RHS) of a differential equation contains only an exponential function. Therefore, the trail solution

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