Question

How to find integrating factor for non exact equations?

Answer 1

Integrating Factors
If a differential equation of the form






is not exact as written, then there exists a function μ( x,y) such that the equivalent equation obtained by multiplying both sides of (*) by μ,







is exact. Such a function μ is called an integrating factor of the original equation and is guaranteed to exist if the given differential equation actually has a solution. Integrating factors turn nonexact equations into exact ones. The question is, how do you find an integrating factor? Two special cases will be considered.


Case 1:
Consider the differential equation M dx + N dy = 0. If this equation is not exact, then M y will not equal N x ; that is, M y – N x ≠ 0. However, if






is a function of x only, let it be denoted by ξ( x). Then







will be an integrating factor of the given differential equation.

Case 2:
Consider the differential equation M dx + N dy = 0. If this equation is not exact, then M y will not equal N x ; that is, M y – N x ≠ 0. = 0. However, if






is a function of y only, let it be denoted by ψ( y). Then







will be an integrating factor of the given differential equation.

Example 1: The equation






is not exact, since




However, note that






is a function of x alone. Therefore, by Case 1,






will be an integrating factor of the differential equation. Multiplying both sides of the given equation by μ = x yields






which is exact because






Solving this equivalent exact equation by the method described in the previous section, M is integrated with respect to x,







and N integrated with respect to y:







(with each “constant” of integration ignored, as usual). These calculations clearly give







as the general solution of the differential equation. Plz see pic