1. Exact differentials and the theory of differential equations Let’s consider the first order differential equation with We can find F(x,y) that generates dF exact differential form We know from exactness test :exactness test if see chapter differentials Solving F(x,y)=const. with respect to y explicit solution of otherwise F(x,y)=const. implicit solution

2. Example: = Exactness test: exact Finding F(x,y):

3. How do we see that solves Implicit differentiation of What to do if inexact Try to find an integrating factor find a function M(x,y) so that exact Integrating factor

4. Finding M looks more complicate than the original problem But sometimes simple ansatz like M=M(x) or M=M(y) or M=M(x/y) works According to exactness test M must fulfill the partial differential equation Example: Exactness test: inexact = Let’s try ansatz M=M(y) in order to find M: with

5. Solution from We see inexact exact with the solutionknown from the prior example Since solves it also solves but