Answered: d² u 5. dt² = n + cos(t)

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

Solve for the general solution of the differential equation:

Expert Solution

Step 1: ''Introduction to the solution"

The given equation is $fraction numerator d squared u over denominator d t squared end fraction plus u equals cos t.......... left parenthesis 1 right parenthesis$

Then, the given equation can be written as : $open parentheses D squared plus 1 close parentheses u equals cos t comma space w h e r e space D identical to fraction numerator d over denominator d t end fraction$

Then, the auxiliary equation is given by $m squared plus 1 equals 0$ that is $m equals i comma negative i$

So, Complementary solution is $y subscript c f end subscript left parenthesis x right parenthesis equals open parentheses A cos t plus B sin t close parentheses comma w h e r e space space A space a n d space B space b e i n g space a r b i t r a r y space c o n s tan t s.$

Let us now find the particular solution of the given differential equation by variation of parameter.