Solve the given symbolic initial value problem and sketch a graph of the solution. y′′+y=3delta(t-(frac{pi}{2}) y(0)=0​ y'(0)=3

sjeikdom0 2020-11-23 Answered
Solve the given symbolic initial value problem and sketch a graph of the solution.
y+y=3δ(t(π2)y(0)=0y(0)=3
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Expert Answer

wornoutwomanC
Answered 2020-11-24 Author has 81 answers
Step 1
Consider the provided question,
Solving the given D.E by using the Laplace Transform.
Given that , y+y=3δ(t(π2)y(0)=0y(0)=3
take the Laplace transform on both side.
L(y")+L(y)=3Lδ(t(π/2))
Since , Lδ(ta)=eas
s2Y(s)sy(0)y(0)+Y(s)=3eπ2s
s2Y(s)03+Y(s)=3eπ2s
Y(s)(s2+1)=3eπ2s+3
Y(s)=3eπ2ss2+1+3s2+1
L(y)=3eπ2s(s2+1)+3(s2+1)
Step 2
Now, use the inverse Laplace transformation,
L(y)=3eπ2s(s2+1)+3(s2+1)
y=L1(3eπ2s(s2+1))+L1(3(s2+1))
Use L1(a(s2+a2))=sin(at)
(since, L1{easF(s)}=H(ta)f(ta))
Where , H(t) is Heaviside step function
y=H(tπ2)3sin(tπ2)+3sin(t)
Thus , solution of the given D.E. is ,
y=3H(tπ2)sin(tπ2)+3sin(t)
Step 3
The graph of the above solution is drawn as,
y=3H(tπ2)sin(tπ2)+3sin(t)
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