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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

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

$ydy-\frac{5}{4}x^2dx=0$
4

The differential equation $ydy-\frac{5}{4}x^2dx=0$ 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 $f(x,y)=C$

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

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

$\frac{d}{dy}\left(-\frac{5}{4}x^2\right)$

The derivative of the constant function ($-\frac{5}{4}x^2$) is equal to zero

0

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

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

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

0
5

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

$0=0$

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 $2$

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

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

$-\frac{5}{12}x^{3}+g(y)$
6

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

$-\frac{5}{12}x^{3}+g(y)$

The derivative of the constant function ($-\frac{5}{12}x^{3}$) is equal to zero

0

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

$0+g'(y)$
7

Now take the partial derivative of $-\frac{5}{12}x^{3}$ with respect to $y$ to get

$0+g'(y)$

Simplify and isolate $g'(y)$

$y=0+g$

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

$y=g$

Rearrange the equation

$g=y$
8

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

$g'(y)=y$

Integrate both sides with respect to $y$

$g=\int ydy$

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 $n=1$

$g=\frac{1}{2}y^2$
9

Find $g(y)$ integrating both sides

$g(y)=\frac{1}{2}y^2$
10

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

$f(x,y)=-\frac{5}{12}x^{3}+\frac{1}{2}y^2$
11

Then, the solution to the differential equation is

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

We need to isolate the dependent variable $y$, we can do that by subtracting $-\frac{5}{12}x^{3}$ from both sides of the equation

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

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

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

Removing the variable's exponent

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

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

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

Multiply the single term $2$ by each term of the polynomial $\left(C_0+\frac{5}{12}x^{3}\right)$

$y=\sqrt{2C_0+\frac{5}{6}x^{3}},\:y=-\sqrt{2\left(C_0+\frac{5}{12}x^{3}\right)}$

We can rename $2C_0$ as other constant

$y=\sqrt{C_0+\frac{5}{6}x^{3}},\:y=-\sqrt{2\left(C_0+\frac{5}{12}x^{3}\right)}$

Multiply the single term $2$ by each term of the polynomial $\left(C_0+\frac{5}{12}x^{3}\right)$

$y=\sqrt{C_0+\frac{5}{6}x^{3}},\:y=-\sqrt{2C_0+\frac{5}{6}x^{3}}$
12

Find the explicit solution to the differential equation

$y=\sqrt{C_0+\frac{5}{6}x^{3}},\:y=-\sqrt{2C_0+\frac{5}{6}x^{3}}$

Final Answer

$y=\sqrt{C_0+\frac{5}{6}x^{3}},\:y=-\sqrt{2C_0+\frac{5}{6}x^{3}}$

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