Definition

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

Solution

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