Question

Find the equation of a curve passing through the point $(0,0)$ and whose differential equation is $y^{\prime }=e^x\sin x$.

Answer 1

Equation is given by $y^{{}^{\prime }}=e^x\sin x$

$dy=e^x\sin xdx$

Integrating both side we get(use integration by parts)

$y=\frac{e^x\cos x+e^x\sin x}{2}+c$

Now, it has been given that this curve will pass from $(0,0)$, so putting this in above equation we will get $c=\frac{-1}{2}$

Hence, equation of the curve will be
$y=\frac{e^x\cos x+e^x\sin x}{2}-\frac{1}{2}$