How to Solve Exact First Order Differential Equations

A differential equation is exact if you can integrate the right-hand side and the left-hand side directly. This is the simplest type of differential equations.

Rule

Exact Differential Equations

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

Example 1

Solve the differential equation $y^{'} = \frac{1}{x} + 2 x$

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

\begin{array}{l} y = \int y^{'} d x = \int \frac{1}{x} + 2 x d x = \ln | x | + x^{2} + C \end{array}

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