# What is a solution to the differential equation ydy/dx=e^x with y(0)=4?

Jul 9, 2016

$\implies {y}^{2} / 2 = {e}^{x} + 7$

or if you like...
$\implies y = \pm \sqrt{2 \left({e}^{x} + 7\right)}$

#### Explanation:

this is separable and has already been separated

so we integrate both side wrt x

$\int y \frac{\mathrm{dy}}{\mathrm{dx}} \setminus \mathrm{dx} = \int {e}^{x} \setminus \mathrm{dx}$

$= \int y \setminus \mathrm{dy} = \int {e}^{x} \setminus \mathrm{dx}$

$\implies {y}^{2} / 2 = {e}^{x} + C$

applying the IV

${4}^{2} / 2 = {e}^{0} + C \implies C = 7$

$\implies {y}^{2} / 2 = {e}^{x} + 7$