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 $

Answered: b) y" + 3y + 2y = e; y(0) = 0, y'(0) =…

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 $

Expert Solution

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.