# Solve the differential equation y dy/dx-y^2+9x=0 ?

Mar 21, 2017

$y = \sqrt{{C}_{0} {e}^{2 x} + 9 \left(x + \frac{1}{2}\right)}$

#### Explanation:

Think that

$y \frac{\mathrm{dy}}{\mathrm{dx}} = \frac{1}{2} \frac{d}{\mathrm{dx}} {y}^{2}$ so the equation can be read as

$\frac{1}{2} \frac{d}{\mathrm{dx}} {y}^{2} - {y}^{2} + 9 x = 0$ now calling $z = {y}^{2}$ we have

$\frac{1}{2} \frac{\mathrm{dz}}{\mathrm{dx}} - z + 9 x = 0$. This equation is easy to solve giving

$z = {C}_{0} {e}^{2 x} + 9 \left(x + \frac{1}{2}\right)$ and finally

$y = \sqrt{z} = \sqrt{{C}_{0} {e}^{2 x} + 9 \left(x + \frac{1}{2}\right)}$