Change of variables

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School:Mathematics > Differential Equations > Ordinary 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

Substitute a term for a variable (e.g. $\textstyle u=\frac {2y}{x}$ ).
Implicitly differentiate the variable (e.g. $\textstyle \frac {du}{dx}=\frac {2}{x} \frac {dy}{dx}$ ).
Solve for the derivative that needs to be solved (e.g. $\textstyle \frac {dy}{dx}=\frac {x}{2} \frac {du}{dx}$ ).
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 $y$ .

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