Solve the following initial value problems using Laplace Transforms: displaystylefrac{{{d}^{2}{y}}}{{{left.{d}{x}right.}^{2}}}+{25}{y}={t} y(0)=0 y'(0)=0.04

Armorikam 2021-02-26 Answered
Solve the following initial value problems using Laplace Transforms:
d2ydx2+25y=t
y(0)=0
y(0)=0.04
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Expert Answer

FieniChoonin
Answered 2021-02-27 Author has 102 answers
Step 1
The given IVP is given as follows.
d2ydx2+25y=t,   y(0)=0,   y(0)=0.04
Apply the laplace transform on both sides of the differential equation as follows.
L{y}+25L{y}=L{t}
s2L{y}sy(0)y(0)+25L{y}=1s2
s2L{y}+25L{y}=1s2+0.04
L{y}=1s2(s2+25)+0.04s2+25
L{y}=125s2125(s2+25)+0.04s2+25
Step 2
Apply inverse laplace transforms on both sides.
y(t)=L1{125s2}L1{125(s2+25)}+L1{0.04s2+25}
=125L1{1s2}1125L1{5s2+52}+0.008L1{5s2+52}
=t251125sin(5t)+0.008sin(5t)
=t25
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