4. Write the general solution of the differential equationdx²dx+ 5y = 2sin²2x-x²5

Given Differential equation is d2ydx2-5dydx+5y=2sin22x-x2-5d2ydx2-5dydx+5y=-5-x2+(1-cos4x)Since…

Answered: d²y 1²2-5x + 5y = 2sin²2x - x² - 5 4.…

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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Related questions

Question

4. Write the general solution of the differential equation
dx²
dx
+ 5y = 2sin²2x-x²5

Expert Solution

Step 1

Given Differential equation is

$\frac{d^{2} y}{d x^{2}} - 5 \frac{d y}{d x} + 5 y = 2 \sin^{2} 2 x - x^{2} - 5 \frac{d^{2} y}{d x^{2}} - 5 \frac{d y}{d x} + 5 y = - 5 - x^{2} + (1 - \cos 4 x) S i n c e 2 \sin^{2} 2 x = 1 - \cos (4 x) \frac{d^{2} y}{d x^{2}} - 5 \frac{d y}{d x} + 5 y = - 4 - x^{2} - \cos (4 x)$

The Auxiliary equation is

$\frac{d^{2} y}{d x^{2}} - 5 \frac{d y}{d x} + 5 y = 0 D^{2} - 5 D + 5 = 0 D = \frac{5 \pm \sqrt{25 - 20}}{2} D = \frac{5 \pm \sqrt{5}}{2} D = \frac{5}{2} \pm \frac{\sqrt{5}}{2} T h e A x u l l a r y s o l u t i o n i s y_{c} = e^{\frac{5 x}{2}} [c_{1} \cos \frac{\sqrt{5} x}{2} + c_{2} \sin \frac{\sqrt{5} x}{2}]$