(a)
Apply Laplace transform on both sides :
Using the formula for Laplace transforms:
=>
=>
=>
=>
=> {Factorizing denominator}
=> { Using partial fractions }
Taking inverse laplace transform:
(b) The given integral equation can also be written as :
=>
Differentiating with respect to t :
=>
Again differentiating with respect to t :
=>
=>
The initial value problem is therefore:
(c) The characteristic equation is given by :
r^{2} -1 =0
=> r^{2} = 1 => r =-1, +1
The general solution is given by :
=>
Putting initial conditions:
Solving these equations , we get :
2A = 1 => A=1/2 and B = 1/2
Hence particular solution is given by :
This is same as we have got in part a)
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