# Step-by-step Solution

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

Go!
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### Videos

$y=\sqrt[3]{x^2+C_0}$

## Step-by-step explanation

Problem to solve:

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

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

$3y^2dy=2xdx$
2

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

$\int3y^2dy=\int2xdx$
3

Solve the integral $\int3y^2dy$ and replace the result in the differential equation

$y^{3}=\int2xdx$
4

Solve the integral $\int2xdx$ and replace the result in the differential equation

$y^{3}=x^2$
5

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

$y^{3}=x^2+C_0$
6

Removing the variable's exponent

$y=\sqrt[3]{x^2+C_0}$

$y=\sqrt[3]{x^2+C_0}$

### Problem Analysis

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

### Main topic:

Differential equations

~ 0.05 seconds