1

Here, we show you a step-by-step solved example of separable differential equations. This solution was automatically generated by our smart calculator:

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

2

Rewrite the differential equation in the standard form $M(x,y)dx+N(x,y)dy=0$

$3y^2dy-2xdx=0$

3

The differential equation $3y^2dy-2xdx=0$ is exact, since it is written in the standard form $M(x,y)dx+N(x,y)dy=0$, where $M(x,y)$ and $N(x,y)$ are the partial derivatives of a two-variable function $f(x,y)$ and they satisfy the test for exactness: $\displaystyle\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}$. In other words, their second partial derivatives are equal. The general solution of the differential equation is of the form $f(x,y)=C$

$3y^2dy-2xdx=0$

** Intermediate steps

Find the derivative of $M(x,y)$ with respect to $y$

$\frac{d}{dy}\left(-2x\right)$

The derivative of the constant function ($-2x$) is equal to zero

Find the derivative of $N(x,y)$ with respect to $x$

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

The derivative of the constant function ($3y^2$) is equal to zero

4

Using the test for exactness, we check that the differential equation is exact

$0=0$

** Intermediate steps

The integral of a function times a constant ($-2$) is equal to the constant times the integral of the function

$-2\int xdx$

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 $n=1$

$-2\cdot \left(\frac{1}{2}\right)x^2$

Multiply the fraction and term in $-2\cdot \left(\frac{1}{2}\right)x^2$

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

Multiply $-2$ times $1$

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

Divide $-2$ by $2$

$-x^2$

Since $y$ is treated as a constant, we add a function of $y$ as constant of integration

$-x^2+g(y)$

5

Integrate $M(x,y)$ with respect to $x$ to get

$-x^2+g(y)$

** Intermediate steps

The derivative of the constant function ($-x^2$) is equal to zero

The derivative of $g(y)$ is $g'(y)$

$0+g'(y)$

6

Now take the partial derivative of $-x^2$ with respect to $y$ to get

$0+g'(y)$

** Intermediate steps

Simplify and isolate $g'(y)$

$3y^2=0+g$

$x+0=x$, where $x$ is any expression

$3y^2=g$

Rearrange the equation

$g=3y^2$

7

Set $3y^2$ and $0+g'(y)$ equal to each other and isolate $g'(y)$

$g'(y)=3y^2$

** Intermediate steps

Integrate both sides with respect to $y$

$g=\int3y^2dy$

The integral of a function times a constant ($3$) is equal to the constant times the integral of the function

$g=3\int y^2dy$

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 $2$

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

Multiplying the fraction by $3$

$g=\frac{3y^{3}}{3}$

Simplify the fraction $\frac{3y^{3}}{3}$ by $3$

$g=y^{3}$

8

Find $g(y)$ integrating both sides

$g(y)=y^{3}$

9

We have found our $f(x,y)$ and it equals

$f(x,y)=-x^2+y^{3}$

10

Then, the solution to the differential equation is

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

** Intermediate steps

Group the terms of the equation

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

Raise both sides of the equation to the exponent $\frac{1}{3}$

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

11

Find the explicit solution to the differential equation. We need to isolate the variable $y$

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

** Final answer to the problem

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