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First order differential equations Calculator

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1

Solved example of first order differential equations

$\frac{dy}{dx}=\frac{5x^2}{4y}$
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

$4ydy=5x^2dx$
3

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

$\int4ydy=\int5x^2dx$

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

$4\int ydy$

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

$2y^2$
4

Solve the integral $\int4ydy$ and replace the result in the differential equation

$2y^2=\int5x^2dx$

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

$5\int x^2dx$

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

$5\frac{x^{3}}{3}$

Simplify the fraction

$\frac{5}{3}x^{3}$
5

Solve the integral $\int5x^2dx$ and replace the result in the differential equation

$2y^2=\frac{5}{3}x^{3}$
6

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

$2y^2=\frac{5}{3}x^{3}+C_0$

Answer

$2y^2=\frac{5}{3}x^{3}+C_0$

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