How to Write and Solve Separable Differential Equations

A separable differential equation is a differential equation where the variables can be separated to either side of the equal sign. This means that you can put all the $x$ -terms on one side and all the $y$ -terms on the other. It is best to collect all the $y$ -terms on the left-hand side and all the $x$ -terms on the right-hand side. Generally, you write separable equations on this form:

Rule

Separable Differential Equations

\begin{array}{l} g (y) y^{'} = f (x) \\ \Rightarrow \int g (y) d y = \int f (x) d x + C \end{array}

g (y) y^{'} = f (x) \Rightarrow \int g (y) d y = \int f (x) d x + C

Example 1

Solve the differential equation $2 x y y^{'} = x^{2}$

\begin{array}{l} 2 x y y^{'} & = x^{2} | \div 2 x \\ y y^{'} & = \frac{x^{2}}{2 x} = \frac{x}{2} \\ \int y d y & = \int \frac{x}{2} d x \\ \frac{1}{2} y^{2} & = \frac{x^{2}}{4} + C \\ y & = \pm \sqrt{\frac{x^{2}}{2} + C} \end{array}

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