What you state is only true of equations withconstant coefficients. It is easy to show that the set of all solutions to an nth order linear, homogeneous, differential equation is a vector space of dimension n. That means that if we have twoindependentsolutions to the equation,anysolution can be written as a linear combination of those two solutions. In particular, if the coefficients are constant, the eigenvalues, and , you give are solutions to the "characteristic equation". If a second order equation has repeated eigenvalues, , that means that the characteristic equation is of the form and so the differential equation is .

X

It is easy to show that satisfies that equation. What about ? What do you get when you put that into the equation? Can you show that those two functions are independent?