Continuous Eigenvalues

0:56 How can I be sure that this continuous sum is the general solution of my D.E. Let say this was a solution of a D.E. Thanks in advance!

[@nicolas_heringer]

Just plug it into your D.E. and see it for yourself. If this not convinced you, just need to think about those psi's beeing a basis of you space, and those phi's beeing the "linear combination weight" of your basis vectors. So as your space is a vector space, than a linear combination must still inside your space, so you can build any function in your space by taking that linear combination of a continuous set of basis function (this is kind of the meaning of the integral).