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www.r X IMI Exam[ X WeB W X c fi X nstruct math Isa umich.edu /webwork2/ma216-f15/homework10/16/?effectiveUser sebojeda&user; sebojeda&theme; &key; MUQSRJuNQ. ABP E Soundcloud YouTube G Gmail a Amazon M Wolverine Access CTools M M Print D Math 216 M Math Hw G ChE 230 Tutoring C Chegg lM Sports M AADL b Slader (0 points) Math 216 Homework homework 10, Problem 16 Use Laplace transforms to find a nontrivial solution to Require that ar(1) e 2 to find any arbitrary constant in your solution (t) 2:33 PM NE P g A Ask me anything 2/11/2015

Explanation / Answer

L{tx''} + 2 L{tx'} 3 L{x'} + 2 X(s) = 0

- (d/ds)[s^2 F(s) - s x(0) - x'(0)] - 2 (d/ds)[s F(s) - x(0)] 3 [s F(s) - x(0)] + 2 F(s) = 0

- (d/ds)[s^2 F(s) - x'(0)] - 2 (d/ds)[s F(s)] 3 [s F(s)] + 2 F(s) = 0, since x(0) = 0.

- [2s F(s) + s^2 F'(s)] - 2 [F(s) + s F'(s)] 3 [s F(s)] + 2 F(s) = 0

(s^2 + 2s) F'(s) + 5s F(s) = 0

F' / F = -5/(s + 2)

ln F = -5 ln(s + 2) + A  

Thus, F(s) = B(s + 2)^(-5)

x(t) = B e^(-2t) * t^4 / 4!

Now, we can find B via x(1) = e^(-2).  

Thus we get B= 4!

putting the value of B, we get x(t)= e^(-2t) * t^4

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