Solve the following differential equations by Laplace transform:d'rdt²given that at t=0, x= 1 and+4drdtdr2dt= 2.+5x = 10e²

Answered: Solve the following differential…

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 the following differential equations by Laplace transform:
d'r
dt²
given that at t=0, x= 1 and
+4
dr
dt
dr
2
dt
= 2.
+5x = 10e²

Expert Solution

Step 1

Sol:-

$x'' (t) + 4 x' + 5 x = 10 e^{t} S o l v e u s i n g L a p l a c e t r a n s f o r m Take Laplace transform of both sides of the equation L \{x'' (t) + 4 x' + 5 x\} = L \{10 e^{t}\} s^{2} L \{x\} - sx (0) - x' (0) + 4 (sL \{x\} - x (0)) + 5 L \{x\} = \frac{10}{s - 1} Plug in the initial conditions : x (0) = 1, x' (0) = 2 s^{2} L \{x\} - s \cdot 1 - 2 + 4 (sL \{x\} - 1) + 5 L \{x\} = \frac{10}{s - 1} Isolate L \{x\} : L \{x\} = \frac{s^{2} + 5 s + 4}{(s - 1) (s^{2} + 4 s + 5)} Take the inverse Laplace transform x = L^{- 1} \{\frac{s^{2} + 5 s + 4}{(s - 1) (s^{2} + 4 s + 5)}\}$