Determine the solution of y(t) for the following equations

Cheyanne Leigh

Cheyanne Leigh

Answered question

2021-09-15

Determine the solution of y(t) for the following equations
dydt+50tydt+2y=10  with  y(0)=0

Answer & Explanation

Viktor Wiley

Viktor Wiley

Skilled2021-09-16Added 84 answers

Step 1 Introduction
For finding the solution of given differential equation we apply Laplace transform method.
Step 2 Step-by-Step Explanation
Given:
dydt+50tydt+2y=10
with y(0)=0
Taking laplace transform on both sides,
L{dydt}+2L{y(t)}+5L{0ty(t)dt}=L{10}
[sL{y(t)}y(0)]+2L{y(t)}+5L{y(t)}s=10s
sL{y(t)}+2L{y(t)}+5L{y(t)}s=10s
L{y(t)}=[s+2+5s]=10s
L{y(t)}=10ss(s(s+2)+5)
L{y(t)}=10s2+2s+5
L{y(t)}=ϕ(s)=10s2+2s+5
by Applying laplace inverse on both sides
y(t)=L1{10s2+2s+5}
=L1{10(s+1)2+4}
=5etL1{2s2+4}
by first shifting property i.e. L1{ϕ(s+a)}=eatf(t)
use L1{as2+a2}=sinat
y(t)=5etsin2t
which is required solution of given Differential equation

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