3. (a) Solve the given differential equationy" + 4y = (x² – 3) sin 2r.(b) Without solving. What is the form of a particular solution ofy" + y = x3 + 2+ cos 2x.

Answered: 3. (a) Solve the given differential…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Concept explainers

Question

3. (a) Solve the given differential equation
y" + 4y = (x² – 3) sin 2r.
(b) Without solving. What is the form of a particular solution of
y" + y = x3 + 2+ cos 2x.

Expert Solution

Solution:

The differential equation is $y'' + 4 y = (x^{2} - 3) \sin 2 x$ .

Obtain the homogeneous solution of $y'' + 4 y = 0$ as $y_{h} (x) = c_{1} \cos (2 x) + c_{2} \sin (2 x)$

Consider the particular solution as $y_{p} (x) = (A x^{2} + B x + C) \sin (2 x)$ .

Then

$y_{p}' (x) = 2 (A x^{2} + B x + C) \cos (2 x) + (2 A x + B) \sin (2 x) y_{p}'' (x) = - 4 (A x^{2} + B x + C) \sin (2 x) + 2 (2 A x + B) \cos (2 x) + 2 (2 A x + B) \cos (2 x) + 2 A \sin (2 x) = [- 4 (A x^{2} + B x + C) + 2 A] \sin (2 x) + 4 (2 A x + B) \cos (2 x)$

Substitute the results:

So we have

$[- 4 (A x^{2} + B x + C) + 2 A] \sin (2 x) + 4 (2 A x + B) \cos (2 x) + 4 (A x^{2} + B x + C) \sin (2 x) = (x^{2} - 3) \sin 2 x 2 A \sin (2 x) = (x^{2} - 3) \sin 2 x A = \frac{- 3}{2}, and we can consider B = 0 a n d C = - 3$

Hence, the particular solution is $y_{p} (x) = (- \frac{3}{2} x^{2} - 3) \sin (2 x)$

Therefore, the general solution is $y (x) = y_{h} (x) + + y_{p} (x) = c_{1} \cos (2 x) + c_{2} \sin (2 x) + (- \frac{3}{2} x^{2} - 3) \sin (2 x)$ .

Step by step

Solved in 3 steps

Knowledge Booster

$Differential Equation$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.

Recommended textbooks for you

Advanced Engineering Mathematics

Advanced Math

ISBN:

9780470458365

Author:

Erwin Kreyszig

Publisher:

Wiley, John & Sons, Incorporated