Differential equations/Homogeneous differential equations

Definition
The word “homogeneous” can mean different things depending on what kind of differential equation you’re working with. A homogeneous equation in this sense is defined as one where the following relationship is true:

$$\textstyle f(tx,ty) = t \cdot f(x,y)$$

Solution
The solution to a homogeneous equation is to:


 * 1) Use the substitution $$\textstyle y = ux$$ where $$u$$ is a substitution variable.
 * 2) Implicitly differentiate the above equation to get $$ \frac {dy} {dx} = x \frac {du} {dx} + u $$.
 * 3) Replace $$\textstyle \frac {dy} {dx}$$ and $$\textstyle y$$ with these expressions.
 * 4) Solve for $$u$$.
 * 5) Substitute with the expression $$ u = \frac {y} {x} $$ Then solve for $$\textstyle y$$.

The advantage of this method is that the function is in terms of 2 variables, but we simplify the equation by relating $$\textstyle y$$ and $$\textstyle x$$ to each other.