Exact Differential Equations

The total differential [Maple Math] of a function [Maple Math] is defined by

[Maple Math] . Suppose that we can write a differential equation as

[Maple Math] . If [Maple Math] is the partial derivative of [Maple Math] with respect to [Maple Math] , and [Maple Math] is the partial derivative of [Maple Math] with respect to [Maple Math] , then [Maple Math] is said to be an exact differential, and the equation [Maple Math] is called an exact differential equation. It is easy to test when an equation is an exact differential equation, since that is the case if and only if

[Maple Math] . Once we check that the equation is exact we use the following formula for [Maple Math] .

[Maple Math] . To faciliate calulations we introduce the following operator

> Ex:=int(M(x,y),x)+int((N(x,y)-int(diff(M(x,y),y),x)),y);

(Here Ex stands for exact.)

[Maple Math]

Example: Solve the equation [Maple Math] .

Example: Solve [Maple Math] .

Often a differential equation of the form [Maple Math] is not exact, but it could be made exact by multiplying both sides with another function [Maple Math] . In general, finding [Maple Math] is very difficult. We will only consider two special cases when the difference [Maple Math] is [Maple Math] times a function of [Maple Math] , or [Maple Math] times a function of [Maple Math] . In the first case we define [Maple Math] , while in the second case we define [Maple Math] . Then

[Maple Math] in the first case, and dth=179 height=98 alt= in the second case.

Example: Solve [Maple Math] .