# Question #3f0a4

Aug 9, 2016

$y = 3 {e}^{\sin} \left(x\right)$

#### Explanation:

Solving the separable differential equation, first we break up the terms into $f \left(x\right) \mathrm{dx} = g \left(y\right) \mathrm{dy}$, and then integrate:

$\frac{\mathrm{dy}}{\mathrm{dx}} = y \cos \left(x\right)$

$\implies \frac{1}{y} \mathrm{dy} = \cos \left(x\right) \mathrm{dx}$

$\implies \int \frac{1}{y} \mathrm{dy} = \int \cos \left(x\right) \mathrm{dx}$

$\implies \ln \left(y\right) = \sin \left(x\right) + C$

At $x = 0 , y = 3$:

$\ln \left(3\right) = \sin \left(0\right) + C$

$\implies C = \ln \left(3\right)$

$\implies \ln \left(y\right) = \sin \left(x\right) + \ln \left(3\right)$

$\therefore y = {e}^{\sin \left(x\right) + \ln \left(3\right)}$

$= {e}^{\ln} \left(3\right) {e}^{\sin} \left(x\right)$

$= 3 {e}^{\sin} \left(x\right)$