Solve the initial value problem using Laplace transforms. y"+y=f(t) , y(0)=0 , y'(0)=1

Lennie Carroll 2021-09-06 Answered

Solve the initial value problem using Laplace transforms.
y"+y=f(t),y(0)=0,y(0)=1
Here
f(t)={00t<3π1t3π

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Expert Answer

Arnold Odonnell
Answered 2021-09-07 Author has 109 answers

Given differential equation
y"+y=f(t),y(0)=0,y(0)=1
Here
f(t)={00t<3π1t3π
To find: Solve the initial value problem using Laplace Transform
We begin with taking Laplace transformation on both sides,
L(yy)=L(f(t))
L(y+L(y)=L(f(t))
Note: L(y=s2L(y)sy(0)y(0)
Substitute in the equation,
(s2L(y)sy(0)y(0))+L(y)=L(f(t))
We begin with finding the Laplace for RHS,
L(f(t))=03πest(0)dt+3πest(1)dt
=3πest(1)dt
=ests3π
=e3πss
Now, use the conditions given,
s2L(y)1+L(y)=L(f(t))
(s2+1)L(y)1=e3πss
(s2+1)L(y)=e3πss+1
L(y)=e3πss(s2+1)+1s2+1
Now , find inverse Laplace transform
y(t)=L1(e3πss(s2+1))+L1(1s2+1)
Note:

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