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Separable differential equations Calculator

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1

Solved example of separable differential equations

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

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

$3y^2dy=2xdx$
3

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

$\int3y^2dy=\int2xdx$

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

$3\int y^2dy$

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

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

Simplify the fraction

$y^{3}$
4

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

$y^{3}=\int2xdx$

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

$x^2$
5

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

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

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

$y^{3}=x^2+C_0$
7

Removing the variable's exponent

$\sqrt[3]{y^{3}}=\sqrt[3]{x^2+C_0}$
8

Applying the power of a power property

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

Answer

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

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