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:

$M(t,y)+N(t,y)y'=0\,$ or $M(t,y)dt+N(t,y)dy=0\,$

(we will call this equation 1)

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

${\frac {\partial M}{\partial y}}={\frac {\partial N}{\partial t}}$

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

To find $y$, manipulate the substitutions of M and N to get $M\partial t=\partial \phi$ and $N\partial y=\partial \phi$. Integrate both sides. This will give us $\phi (t)\,$ and $\phi (y)\,$. To get $\phi (t,y)\,$, 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 $y$, use the quadratic formula.