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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

Take $\frac{5}{4}$ out of the fraction

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

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

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

$\int ydy=\int\frac{5}{4}x^2dx$

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 $n=1$

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

Solve the integral $\int ydy$ and replace the result in the differential equation

$\frac{1}{2}y^2=\int\frac{5}{4}x^2dx$

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

$\frac{5}{4}\int x^2dx$

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 $2$

$\frac{5}{4}\left(\frac{x^{3}}{3}\right)$

Simplify the fraction $\frac{5}{4}\left(\frac{x^{3}}{3}\right)$

$\frac{5}{12}x^{3}$
6

Solve the integral $\int\frac{5}{4}x^2dx$ and replace the result in the differential equation

$\frac{1}{2}y^2=\frac{5}{12}x^{3}$
7

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

$\frac{1}{2}y^2=\frac{5}{12}x^{3}+C_0$
8

Eliminate the $\frac{1}{2}$ from the left, multiplying both sides of the equation by $$

$y^2=2\left(\frac{5}{12}x^{3}+C_0\right)$
9

Removing the variable's exponent

$y=\pm \sqrt{2\left(\frac{5}{12}x^{3}+C_0\right)}$
10

As in the equation we have the sign $\pm$, this produces two identical equations that differ in the sign of the term $\sqrt{2\left(\frac{5}{12}x^{3}+C_0\right)}$. We write and solve both equations, one taking the positive sign, and the other taking the negative sign

$y=\sqrt{2\left(\frac{5}{12}x^{3}+C_0\right)},\:y=-\sqrt{2\left(\frac{5}{12}x^{3}+C_0\right)}$

Final Answer

$y=\sqrt{2\left(\frac{5}{12}x^{3}+C_0\right)},\:y=-\sqrt{2\left(\frac{5}{12}x^{3}+C_0\right)}$

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