Exact differential equation

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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.

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