Exact Equations

Author: John J Weber III, PhD Corresponding Textbook Sections:

Expected Educational Results

Exact Equations

Exact Differential Form of an ODE

We will use another notation for total differential in this course. Since F(x,y)=C, then dF=0 and Fxdx+Fydy=0

or

M(x,y)dx+N(x,y)dy=0

In reality, the DE will more often be in the form:

dydx=M(x,y)N(x,y)

Definition: Exact Differential Form

The differential form M(x,y)dx+N(x,y)dy=0 is exact in a rectangle R if there is a function F(x,y) such that F(x,y)x=M(x,y) and F(x,y)y=N(x,y) for all (x,y) in R.

Definition: Exact Equation

If M(x,y)dx+N(x,y)dy=0 is in exact differential form, then the DE is called an exact equation.

Test for Exactness

Suppose M(x,y) and N(x,y) are continuous in a rectangle R, then M(x,y)dx+N(x,y)dy=0 is an exact equation in R if and only if M(x,y)y=N(x,y)x holds for all (x,y) in R.

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Last Modified: Wednesday, 2 September 2020 07:42 EDT