Exact differential equations

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School:Mathematics > Topic:Differential_Equations > Ordinary Differential Equations > Exact Differential Equations

Definition

A differential equation of is said to be exact if it can be written in the form $M(x,y) dx + N(x,y) dy = 0$ where $M$ and $N$ have continuous partial derivatives such that $\frac {\partial M}{\partial y} = \frac {\partial N}{\partial x}$.

Solution

Solving the differential equation consists of the following steps:

1. Create a function $f(x,y) := \int M(x,y) dx$. While integrating, add a constant function $g(y)$ that is a function of $y$. This is a term that becomes zero if function $f(x,y)$ is differentiated with respect to $x$.
2. Differentiate the function $f(x,y)$ with respect to $\frac {\partial f}{\partial y}$. Set $\frac {\partial f}{\partial y} = N(x,y)$. Solve for the function $g(y)$.