Can an Exponential Factor Create a New Solution for a Second Order ODE?

[intervoxel]

Is it possible to form a new solution of a second order ODE by multiplying it by an exponential factor?

 

[g_edgar]

Not clear what you mean. In a second-order LINEAR diff eq, if you have one solution u(x) you can find the general solution by trying the form f(x)u(x) ... This should be in standard ODE textbooks.

 

[intervoxel]

What I mean is that I have two particular solutions even and odd that diverge at infinity, but I noted that if I multiply them by exp(-z^2 / 4) they behave properly. I'm trying to justify this procedure. Substituting the product of each back into the differential equation doesn't seem to work.

 

[HallsofIvy]

In general, no. Multiplying a solution to a d.e. by another function, exponential or not, does NOT give a new solution.