Question

Find the solution of the differential equation 3e^(3x)dy/dx=â9x/y^2

Answer 1

We want to solve
$3e^{3x} \frac{dy}{dx} = \frac{9x}{y^{2}}$

The ODE is separable into the form
$3y^{2} dy = 9xe^{-3x} \\\\ y^{2} dy = 3xe^{-3x}dx$

Integrate.
$\frac{1}{3} \int y^{2} dy = \int x e^{-3x} dx \\\\ \frac{y^{3}}{9} = -\frac{xe^{-3x}}{3} + \frac{1}{3} \int e^{-3x} dx \\\\ = - \frac{x}{3}e^{-3x} - \frac{1}{9} e^{-3x} + c$

$y^{3} = -Ce^{-3x} (1+3x) \\\\ y = - ce^{-x} \sqrt[3]{1+3x}$

Answer: $y = -ce^{-x} \sqrt[3]{1+3x}, \,\,\, c=constant$