Step-by-step Solution

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

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

Step-by-step Solution

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 $x$

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

The integral of a constant by a function is equal to the constant multiplied by the integral of the function

$\frac{3}{2}\int y^2dy$

Apply the power rule for integration, $\displaystyle\int x^n dx=\frac{x^{n+1}}{n+1}$, where $n$ represents a number or constant function, such as $2$

$\frac{3}{2}\left(\frac{y^{3}}{3}\right)$

Simplify the fraction $\frac{3}{2}\left(\frac{y^{3}}{3}\right)$

$\frac{1}{2}y^{3}$
5

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

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

Applying the power rule for integration, $\displaystyle\int x^n dx=\frac{x^{n+1}}{n+1}$, where $n$ represents a number or constant function, in this case $n=1$

$\frac{1}{2}x^2$
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

Cancel $\frac{1}{2}$ from both sides of the equation

$y^{3}=x^2$
8

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

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

Removing the variable's exponent

$y=\sqrt[3]{x^2+C_0}$
9

Find the explicit solution to the differential equation

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

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