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b) y" + 3y + 2y = e; y(0) = 0, y'(0) = 3 c) y" +9y= sin(2x); y(0) = 1, y'(0) = 0

b) y" + 3y + 2y = e; y(0) = 0, y'(0) = 3 c) y" +9y= sin(2x); y(0) = 1, y'(0) = 0

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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I only need help with parts b and c. Please show all steps clearly

**Problem 4: Initial Value Problems** Solve the following initial value problems: a) \( y'' + 4y = 2x; \quad y(0) = 1, \ y'(0) = 2 \) b) \( y'' + 3y' + 2y = e^x; \quad y(0) = 0, \ y'(0) = 3 \) c) \( y'' + 9y = \sin(2x); \quad y(0) = 1, \ y'(0) = 0 \) d) \( y^{(4)} - y = 5; \quad y(0) = y'(0) = y''(0) = y^{(3)}(0) = 0 \)
Transcribed Image Text:**Problem 4: Initial Value Problems** Solve the following initial value problems: a) \( y'' + 4y = 2x; \quad y(0) = 1, \ y'(0) = 2 \) b) \( y'' + 3y' + 2y = e^x; \quad y(0) = 0, \ y'(0) = 3 \) c) \( y'' + 9y = \sin(2x); \quad y(0) = 1, \ y'(0) = 0 \) d) \( y^{(4)} - y = 5; \quad y(0) = y'(0) = y''(0) = y^{(3)}(0) = 0 \)
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Step 1: Given:

b right parenthesis space space y apostrophe apostrophe plus 3 y apostrophe plus 2 y equals e to the power of x semicolon space y open parentheses 0 close parentheses equals 0 comma space y apostrophe open parentheses 0 close parentheses equals negative 3 c right parenthesis space space y apostrophe apostrophe plus 9 y equals sin open parentheses 2 x close parentheses space semicolon space space space y open parentheses 0 close parentheses equals 1 comma space y apostrophe open parentheses 0 close parentheses equals 0

We have to solve the given differential equation.

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