Question #eebbd

Jan 9, 2017

$y = {e}^{\frac{1}{2} \left({x}^{2} - 1\right) {\sin}^{2} \left({x}^{2}\right)}$

Explanation:

This differential equation is separable. So

$\frac{\mathrm{dy}}{y} = {x}^{3} \cos \left({x}^{2}\right) \mathrm{dx}$. Integrating we have

$\log y = \frac{1}{2} \left(\cos \left({x}^{2}\right) + {x}^{2} \sin \left({x}^{2}\right)\right) + C$ or

$y = {C}_{1} {e}^{\frac{1}{2} \left(\cos \left({x}^{2}\right) + {x}^{2} \sin \left({x}^{2}\right)\right)}$

The initial condition dictates

$1 = {C}_{1} {e}^{\frac{1}{2}}$ so the final solution is

$y = {e}^{\frac{1}{2} \left({x}^{2} - 1\right) {\sin}^{2} \left({x}^{2}\right)}$