My friend asked me this question:
If $y(x)= \int_{0}^{x}f(t)\sin{(px-pt)}dt$ then what is the value of $y''(x)-((p^2)*y(x))$ ?
He gave me the hint to consider $\sin(px-pt)$ as the imaginary part of the complex number $e^{i(px-pt)}$ but this just confused me only more. Please help.
Write $$\sin(px-pt)=\sin(px)\cos(pt)-\cos(px)\sin(pt).$$ Then you can rewrite the function as $$ y(x)=\sin(px)\int_0^xf(t)\cos(pt)dt - \cos(px)\int_0^xf(t)\sin(pt)dt.$$ Now use the product rule and fundamental theorem of calculus to find the derivative and se