Solution the following initial value problem usingthe laplace transformy"+4y=4t, yl0)=1 and yulo)=J

Answered: Solution the following initial Value…

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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Question

Solution the following initial value problem using
the laplace transform
y"+4y=4t
, yl0)=1 and yulo)=J

Expert Solution

Step 1

Given the initial value problem

y''+4y = 4t, y(0)=1, y'(0)=5

Solve it using Laplace transform.

Step 2

Transform both sides of the differential equation.

$L \{y'' + 4 y\} = 4 L \{t\}$

Since the operator $L$ is linear,

$L \{y''\} + 4 L \{y\} = 4 L \{t\} (s^{2} Y (s) - s y (0) - y' (0)) + 4 Y (s) = \frac{4}{s^{2}}$

Step 3

Plug the initial conditions y(0)=1 and y'(0)=5.

$(s^{2} Y (s) - s - 5) + 4 Y (s) = \frac{4}{s^{2}} \Rightarrow (s^{2} + 4) Y (s) = \frac{4}{s^{2}} + s + 5 \Rightarrow Y (s) = \frac{4}{s^{2} (s^{2} + 4)} + \frac{s + 5}{s^{2} + 4} = \frac{4 + s^{2} (s + 5)}{s^{2} (s^{2} + 4)} = \frac{s^{3} + 5 s^{2} + 4}{s^{2} (s^{2} + 4)}$

Step 4

By using partial fraction,

$\frac{s^{3} + 5 s^{2} + 4}{s^{2} (s^{2} + 4)} = \frac{A}{s} + \frac{B}{s^{2}} + \frac{C s + D}{s^{2} + 4} = \frac{A s (s^{2} + 4) + B (s^{2} + 4) + (C s + D) s^{2}}{s^{2} (s^{2} + 4)}$

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