Assume that the equation
,
is not exact, that is-
In this case we look for a function u(x,y) which makes the new equation
,
an exact one. The function u(x,y) (if it exists) is called the integrating factor. Note that u(x,y) satisfies the following equation:
This is not an ordinary differential equation since it involves more than one variable. This is what's called a partial differential equation. These types of equations are very difficult to solve, which explains why the determination of the integrating factor is extremely difficult except for the following two special cases:
,
is a function of x only, that is, the variable y disappears from the expression. In this case, the function u is given by
,
is a function of y only, that is, the variable x disappears from the expression. In this case, the function u is given by
Let us summarize the above technique. Consider the equation
If your equation is not given in this form you should rewrite it first.
,
then compare them.
If this expression is a function of x only, then go to step 3. Otherwise, evaluate
If this expression is a function of y only, then go to step 3.
Otherwise, you can not solve the equation using the technique
developped above!
;
The following example illustrates the use of the integrating factor technique:
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Author: Mohamed Amine Khamsi