Find the Wronskian?

Question

Find the Wronskian of two solutions of t^2*y"-t(t+2)y'+(t+2)y=0 without solving the equation.

Answer: ct^2*e^t

Answer 1

Solve for y'',

y'' = [(t+2)/t] y' - [(t+2)/t^2] y ......(1)

Let y1 and y2 be the two solutions.

W = y1y2' - y1'y2

W' = y1y2'' - y1''y2

Plug in (1),

W' = y1{[(t+2)/t] y2' - [(t+2)/t^2] y2} - {[(t+2)/t] y1' - [(t+2)/t^2] y1} y2

Simplify,

W' = [(t+2)/t]W

Separate variables,

dW/W = (t+2)/t dt

Integrate,

lnW = t + 2lnt + c

W = Ct^2 e^t, where C = e^c