# Step-by-step Solution

## Solve the differential equation $\frac{dy}{dx}=\frac{2x}{3y^2}$

Go!
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### Videos

$y=\sqrt{x^2+2C_0}$

## Step-by-step explanation

Problem to solve:

$\frac{dy}{dx}=\frac{2x}{3y^2}$
1

Take $\frac{2}{3}$ out of the fraction

$\frac{dy}{dx}=\frac{\frac{2}{3}x}{y^2}$
2

Simplify the fraction $\frac{\frac{2}{3}x}{y^2}$

$\frac{dy}{dx}=\frac{x}{\frac{3}{2}y^2}$
3

Group the terms of the differential equation. Move the terms of the $y$ variable to the left side, and the terms of the $x$ variable to the right side

$\frac{3}{2}y^2dy=x\cdot dx$
4

Integrate both sides of the differential equation, the left side with respect to $y$, and the right side with respect to $x$

$\int\frac{3}{2}y^2dy=\int xdx$
5

Solve the integral $\int\frac{3}{2}y^2dy$ and replace the result in the differential equation

$\frac{1}{2}y^{3}=\int xdx$
6

Solve the integral $\int xdx$ and replace the result in the differential equation

$\frac{1}{2}y^{3}=\frac{1}{2}x^2$
7

As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$

$\frac{1}{2}y^{3}=\frac{1}{2}x^2+C_0$
8

Eliminate the $\frac{1}{2}$ from the left, multiplying both sides of the equation by 

$y^{3}=2\left(\frac{1}{2}x^2+C_0\right)$
9

Removing the variable's exponent

$y=\sqrt{2\left(\frac{1}{2}x^2+C_0\right)}$
10

Solve the product $2\left(\frac{1}{2}x^2+C_0\right)$

$y=\sqrt{x^2+2C_0}$

$y=\sqrt{x^2+2C_0}$
$\frac{dy}{dx}=\frac{2x}{3y^2}$