Separable differential equations Calculator

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Difficult Problems

1

Solved example of separable differential equations

$\frac{dy}{dx}=\frac{2x}{3y^2}$
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 $f(x,y)=C$

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

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

0

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

0
4

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

$0=0$

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 $n=1$

$-x^2$

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

$-x^2+g(y)$

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

0

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

$0+g'(y)$

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 $g'(y)$

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

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 $2$

$g=y^{3}$
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

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

We need to isolate the dependent variable $y$, we can do that by subtracting $-x^2$ from both sides of the equation

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

Removing the variable's exponent

$y=\sqrt[3]{C_0+x^2}$
11

Find the explicit solution to the differential equation

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

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