# Differential equations/Change of variables

 Educational level: this is a tertiary (university) resource.
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 Subject classification: this is a mathematics resource.
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### 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 ${\displaystyle y}$ .