Question

Using separation of variables technique, solve the following differential equation with initial condition y'= e sinx and y(pi) = 0. The solution is: ​

Answer 1

$\begin{aligned}&y'=e^y\sin x\\&\dfrac{dy}{dx}=e^y\sin x\\&\dfrac{dy}{e^y}=\sin x\, dx\\&\int \dfrac{dy}{e^y}=\int \sin x\, dx\\&-e^{-y}=-\cos x+C\\&e^{-y}=\cos x+C\\\end$

$e^{-0}=\cos \pi +C\\1=1+C\\C=0\\\\\boxed{\boxed{e^{-y}=\cos x}}$