Using Convolution, Solve the Volterra Integral Equation: y(x)=1-e^{-x}+int_0^x y(x-t)sin x dt

melodykap 2021-01-17 Answered
Using Convolution, Solve the Volterra Integral Equation:
y(x)=1ex+0xy(xt)sinx dt
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Expert Answer

unett
Answered 2021-01-18 Author has 119 answers
Step 1
Given:
y(x)=1ex+0xy(xt)sinx dt
taking Laplace transformation 
Y(s)=1s1s+1+L{0xy(xt)sinx dt}
Y(s)=1s1s+1+Y(s)L(sinx)
Y(s)=1s1s+1+Y(s)11+s2
Y(s)=(11s2+1)=1s(s+1)
Y(s)=s2+1s3(s+1)=(s+1)22ss3(s+1)
Y(s)=s+1s32s2(s+1)
Y(s)=1s2+1s32s2(s+1)
Step 2
Taking Inverse Laplace Transformation
y(x)=L1{1s2}+L1{2s2+1}2L1{1s2(s+1)}
=x+12x2+0x0x(exdx)dx
=x+12x2+0x(ex1)dx
=x+12x2[ex+x]0x
=x+12x2exx+1
y(x)=x+12x2ex
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