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Homogeneous Differential Equation Calculator

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1

Solved example of homogeneous differential equation

$\left(x-y\right)dx+xdy=0$
2

We can identify that the differential equation $\left(x-y\right)dx+x\cdot dy=0$ is homogeneous, 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 both are homogeneous functions of the same degree

$\left(x-y\right)dx+x\cdot dy=0$
3

Use the substitution: $y=ux$

$\left(x-ux\right)dx+x\left(u\cdot dx+x\cdot du\right)=0$

Multiply the single term $x$ by each term of the polynomial $\left(u\cdot dx+x\cdot du\right)$

$\left(x-ux\right)dx+u\cdot x\cdot dx+x^2du=0$

Factor the polynomial $\left(x-ux\right)$ by it's greatest common factor (GCF): $x$

$x\left(1-u\right)dx+u\cdot x\cdot dx+x^2du=0$

Factor by $dx$

$\left(x\left(1-u\right)+ux\right)dx+x^2du=0$

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

$\left(x\left(1-u\right)+ux\right)dx=-x^2du$

Solve the product $x\left(1-u\right)$

$\left(x-xu+ux\right)dx=-x^2du$

Cancel like terms $-xu$ and $ux$

$x\cdot dx=-x^2du$
4

Expand and simplify

$x\cdot dx=-x^2du$
5

Group the terms of the differential equation. Move the terms of the $u$ variable to the left side, and the terms of the $x$ variable to the right side of the equality

$du=\frac{x}{-x^2}dx$

Simplify the fraction by $x$

$\frac{1}{-x}dx$
6

Simplify the expression $\frac{x}{-x^2}dx$

$du=\frac{1}{-x}dx$
7

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

$\int1du=\int\frac{1}{-x}dx$

The integral of a constant is equal to the constant times the integral's variable

$u$
8

Solve the integral $\int1du$ and replace the result in the differential equation

$u=\int\frac{1}{-x}dx$

Take the constant $\frac{1}{-1}$ out of the integral

$-\int\frac{1}{x}dx$

The integral of the inverse of the lineal function is given by the following formula, $\displaystyle\int\frac{1}{x}dx=\ln(x)$

$-\ln\left(x\right)$

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

$-\ln\left(x\right)+C_0$
9

Solve the integral $\int\frac{1}{-x}dx$ and replace the result in the differential equation

$u=-\ln\left(x\right)+C_0$
10

Replace $u$ with the value $\frac{y}{x}$

$\frac{y}{x}=-\ln\left(x\right)+C_0$

Multiply both sides of the equation by $x$

$y=\left(-\ln\left(x\right)+C_0\right)x$
11

Find the explicit solution to the differential equation. We need to isolate the variable $y$

$y=\left(-\ln\left(x\right)+C_0\right)x$

Final Answer

$y=\left(-\ln\left(x\right)+C_0\right)x$

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