# What is a general solution to the differential equation dy/dx=x^3/y^2?

Sep 16, 2016

$y = \sqrt[3]{\frac{3 {x}^{4}}{4} + C}$

#### Explanation:

Treat $\frac{\mathrm{dy}}{\mathrm{dx}}$ like a fraction to move the $y$ terms to one side of the equation and the $x$ terms to the other:

$\frac{\mathrm{dy}}{\mathrm{dx}} = {x}^{3} / {y}^{2} \text{ "=>" } {y}^{2} \mathrm{dy} = {x}^{3} \mathrm{dx}$

Integrate both sides:

$\implies \text{ } \int {y}^{2} \mathrm{dy} = \int {x}^{3} \mathrm{dx}$

Using the typical integration power rule:

$\implies \text{ } {y}^{3} / 3 = {x}^{4} / 4 + C$

Solving for $y$:

$\implies \text{ } {y}^{3} = \frac{3 {x}^{4}}{4} + C$

$\implies \text{ } y = \sqrt[3]{\frac{3 {x}^{4}}{4} + C}$