The following question was given to as in a recent assignment, namely
Let $$y''+p(t)y'+q(t)y=0~.$$ It is given that $(1+t)^2$ is a solution to this differential equation, and that the wronskian of any two solutions is constant. Find the coefficients of $p$ and $q$.
Say for example the wronskian is a constant, say $C$. Then if $u$ is another solution, then $$u'+ \frac{2u}{(1+t)^2} = C~,$$ from which I get $$u = \frac{Ct+A}{(1+t)^2}~,$$ where $A$ is another constant.
How do I obtain the coefficients $p$ and $q$ ? I could solve simultaneous equations, but then they are messy, as the second derivative of $q$ would involve many terms.
How can Abel's identity for differential equations help me? Another problem is that I don't know if the wronskian is $0$ or not.
But $p$ and $q$ are analytic, so if their wronskian is zero then the two functions are linearly dependent; just putting some facts together.
Thanks, Ben