Solved example of homogeneous differential equation
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
Use the substitution: $y=ux$
Multiply the single term $x$ by each term of the polynomial $\left(u\cdot dx+x\cdot du\right)$
Factor the polynomial $\left(x-ux\right)$ by it's greatest common factor (GCF): $x$
Factor by $dx$
We need to isolate the dependent variable $u$, we can do that by simultaneously subtracting $x^2du$ from both sides of the equation
Solve the product $x\left(1-u\right)$
Cancel like terms $-xu$ and $ux$
Expand and simplify
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
Simplify the fraction by $x$
Simplify the expression $\frac{x}{-x^2}dx$
Integrate both sides of the differential equation, the left side with respect to $y$, and the right side with respect to $x$
The integral of a constant is equal to the constant times the integral's variable
Solve the integral $\int1du$ and replace the result in the differential equation
Take the constant $\frac{1}{-1}$ out of the integral
The integral of the inverse of the lineal function is given by the following formula, $\displaystyle\int\frac{1}{x}dx=\ln(x)$
As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$
Solve the integral $\int\frac{1}{-x}dx$ and replace the result in the differential equation
Replace $u$ with the value $\frac{y}{x}$
Multiply both sides of the equation by $x$
Find the explicit solution to the differential equation. We need to isolate the variable $y$
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