Differential equations/Change of variables

Definition
In a differential equation, if a certain term appears many different times, a substitution can be made similar to a $$u$$-substitution.

Solution

 * 1) Substitute a term for a variable (e.g. $$\textstyle u=\frac {2y}{x}$$).
 * 2) Implicitly differentiate the variable (e.g. $$\textstyle \frac {du}{dx}=\frac {2}{x} \frac {dy}{dx}$$).
 * 3) Solve for the derivative that needs to be solved (e.g. $$\textstyle \frac {dy}{dx}=\frac {x}{2} \frac {du}{dx}$$).
 * 4) Solve the original equation in terms of $$u$$ and then use the substitution for $$u$$ to get the original equation back in terms of $$x$$ and $$y$$.