I have been studying differential equations, and developed a theorem about linear second order equations. I have not been able to find it anywhere else on the web or in my textbooks, and so I thought I would post it up here. Click "read more" to see the theorem.

 

If y is the solution of a linear second order differential equation, y'' + py' + qy = g, and assuming p, q, and g are functions of an independent variable, and if q =1/2(p' + 1/2*p^2) then there exists an integrating factor μ = exp(int(p/2)), such that (μy)'' = L[y].

Can anybody give me some feedback? Has anone seen this before in a textbook? Granted it is rare chance that such a μ will ever exst, and it only continues to get "more rare" as the order of the differential equation increases. If one wanted to develop a general theorem stating what the coefficient function must equal in terms of all other constant coefficient functions, then it could done using the binomial theorem, which I could show if anyone is interested. Please give me your feedback, if you have any.