Changes

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

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:

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>

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

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 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.

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

[[Category:Calculus]]

[[Category:Calculus]]

[[Category:Differential Equations]]

[[Category:Differential Equations]]