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solve y′′′′+ 3y′′−4y = t^2

solve y′′′′+ 3y′′−4y = t^2

Advanced Engineering Mathematics
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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solve y′′′′+ 3y′′−4y = t^2

Expert Solution
Step 1

Given differential equation is yIV+3y''−4y=t2.

We have to find the solution of the given differential equation.

Write the auxiliary equation, which is given below

m4+3m2−4=0

Now, simplify m4+3m2−4=0

Rewrite the equation with u=m2 and u2=m4, we get

u2+3u−4=0

⇒u−1u+4=0⇒u=1, −4

Hence, u=1 and u=−4

Substitute back u=m2 and solve for m

m2=1⇒m=1,1

And 

m2=−4⇒m=±2i

Hence, m=1, 1, 2i, -2i

Therefore, the complementary function is given by

yc=c1+c2tet+c3cos2t+c4sin2t.

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