IE1

Solve $y''(x)=f(x), y(0)=A, y'(0)=B\,$

Integrate both sides with respect x.

$y'(x)-y'(0)=\int_0^xf(s)\,ds\,$

$y'(x)-B=\int_0^xf(s)\,ds\,$

$y(x)-y(0)=B\int_0^x\,du+\int_0^x\int_0^uf(s)\,ds\,du\,$

$y(x)=y(0)+Bx+\int_0^x\int_0^uf(s)\,ds\,du\,$

$y(x)=A+Bx + \int_0^x\int_0^uf(s)\,ds\,du\,$

$y(x)=A+Bx + \int_0^x f(s)(x-s)\,ds\,$