Exact differentials and the theory of differential equations

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

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

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

4. = According to exactness test M must fulfill the partial differential equation 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 Example: Exactness test: inexact Let’s try ansatz M=M(y) in order to find M: with

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