Why Are u and v Functions Zero in VOP for Homogeneous 2nd Order Equations?

[lonewolf219]

I just realized you can use variation of parameters (VOP) to solve for homogeneous 2nd order equations. I see it takes much longer to do so. But I was wondering why, if you use VOP, the u and v functions are 0. Is this because the coefficients of the homogeneous equation are constant, or possibly because the differential equation is equal to 0 to begin with?

 

[obafgkmrns]

One way to look at it is simply to note that VOP results in a valid solution to the problem. The uniqueness theorem assures you that you have the only correct solution.