Find the solution of the given initial value problem.

Question

Find the solution of the given initial value problem.

y⁽⁴⁾ + 10y''' + 39y'' + 70y' + 50y = 0;

y(0) = 5, y'(0) = 0, y''(0) = -36, y'''(0) = 190

y(t) =

Answer 1

Given equation is y'''' + 10y''' + 39y'' + 70y' + 50 y = 0

Initial conditions: y(0) = 5, y'(0) = 0, y''(0) = -36, y'''(0) = 190

Take Laplace transform of given equation

s^4 L[y] – s^3 y(0) – s^2y’(0) – sy’’(0) – y’’’(0) + 10 [s^3 L[y] – s^2y’(0) – sy’(0) -y’’(0)]

+39 [ s^2L[y] – sy(0) – y’(0)] +70 [sL[y] – y(0)] + 50 L[y] = 0 Gather like terms

L[y] (s^4+10s^3 + 39s^2 +70s + 50)–y(0) [s^3+10s^2 + 39s + 70) -y’(0) (s^2+10s + 39)-

y”(0) (s + 10) – y’’’(0) (1) = 0          Substitute initial conditions 

L[y] (s^4 + 10s^3 + 39s^2 + 70s + 50) – 5 [s^3 + 10s^2 + 39s + 70) – 0 (s^2 + 10s + 39)

–(-36) (s+10) – 190 = 0 Simplify

L[y] (s^4 + 10s^3 + 39s^2 + 70s + 50) – 5s^3 -50 s^2 -195 s + 36s -350 +360 -190 = 0

L[y] (s^4 + 10s^3 + 39s^2 + 70s + 50) – 5s^3 -50 s^2 – 159s -180 = 0 Rearrange

L[y] = (5s^3 + 50 s^2 +159s +180)/(s^4 + 10s^3 + 39s^2 + 70s + 50)

Use Calculator and find Laplace Inverse to find y(t)

Calculator Link: For calculator link email me at [email protected]



y(t) = 2(3 sin (t) + cos (t)) e^-2t + (7sin (t) + 3 cos (t) ) e^-3t

I hope this helps.

Shailesh (Sky) Kadakia, Expert tutor

WYZANT, Inc.