Differential equations/Change of variables

Educational level: this is a tertiary (university) resource.

Type classification: this is a lesson resource.

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 $u$-substitution.

Solution

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