What is solution of the integral equation y(x)=x^3+integral of sin(x-t)y(t)dt from 0 to x?

Question : What is solution of the integral equation\[y\left(x\right)=x^3+\int_{0}^{x}\sin{\left(x-t\right)}y\left(t\right)dt\ ,\ x\in[0,\ \pi]\]Solution : Given integral equation\[y\left(x\right)=x^3+\int_{0}^{x}\sin{\left(x-t\right)}y\left(t\right)dt\ ,\ x\in[0,\ \pi]\]Can be written as \[y\left(x\right)=x^3+k\left(x\right)\ast y(x)\]Where k(x)*v(x) is convolution of u and v functionsNow applying Laplace transform both sides\[Y\left(s\right)=\frac{6}{s^4}+K\left(s\right)\ Y(s)\]Where K(s) =Laplace…

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