With the aid of Laplace Transform, solve the Initial Value Problem {y}text{}{left({t}right)}-{y}'{left({t}right)}={e}^{t} cos{{left({t}right)}}+ sin{{left({t}right)}},{y}{left({0}right)}={0},{y}'{left({0}right)}={0}

Tahmid Knox 2020-12-09 Answered
With the aid of Laplace Transform, solve the Initial Value Problem
y(t)y(t)=etcos(t)+sin(t),y(0)=0,y(0)=0
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Expert Answer

escumantsu
Answered 2020-12-10 Author has 98 answers
Step 1
The Laplace Transform of a second order linear differential equation ,
y"+ay+by=f(t) with y(0)=A and y(0)=B
is given by,
Y(s)=(s+a)A+Bs2+as+b+F(s)s2+as+b
Given,
y(t)y(t)=etcos(t)+sin(t),y(0)=0,y(0)=0
Step 2
Applying Laplace Transform in the overall equation (1),
y(t)y(t)=etcos(t)+sin(t)
L[y(t),s]L[y(t),s]=L[etcos(t),s]+L[sin(t),s]
[s2Y(s)sy(0)y(0)][sY(s)y(0)]=L[cos(t),s]ss1+as2+12
s2YsY(s)=[ss2+1]ss1+1s2+1
(s2s)Y(s)=[s1(s1)2+1]+1s2+1
s(s1)Y(s)=[s1(s1)2+1]+1s2+1
Y(s)=1s[(s1)2+1]+1s(s1)(s2+1)
Step 3
The answer is :Y(s)=1s[(s1)2+1]+1s(s1)(s2+1)
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