Exact differential equation

An exact differential equation is a differential equation that can be solved in the following manner.

Suppose you are given an equation of the form:

<math>M(t,y) + N(t,y)y' = 0\,</math> or <math>M(t,y) dt + N(t,y) dy = 0\,</math>

(we will call this equation 1)

Before we begin solving it, we must first check that the equation is exact. This means that:

<math>\frac{\partial M}{\partial y} = \frac{\partial N}{\partial t}</math>

To find the solution of this equation, we assume that the solution is φ = constant. We assume the substitution <math>\frac{\partial \phi}{\partial t} = M</math> and <math>\frac{\partial \phi}{\partial y} = N</math>. (If we substitute M and N back into (1), it yields <math>(\frac{\partial \phi}{\partial t}) dt + (\frac{\partial \phi}{\partial y}) dy = 0</math>, which makes sense.)

To find <math>y</math>, manipulate the substitutions of M and N to get <math>M \partial t = \partial \phi</math> and <math>N \partial y = \partial \phi</math>. Integrate both sides. This will give us <math>\phi(t)\,</math> and <math>\phi(y)\,</math>. To get <math>\phi(t, y)\,</math>, write the sum of each term found in each equation. For terms that appear in both equations, only write them once.

To solve the expression for <math>y</math>, use the quadratic formula.