Wronskian and Linear Independence of y1 = t2 and y2 = t|t| in Second Order ODEs

[Gear300]

For y₁ = t² and y₂ = t|t| (y₂'' is not defined at t = 0), the Wronskian is 0 for all t over the interval [-1,1]. However, the two functions are not linearly dependent over this interval in the sense that one is not a unique multiple of the other. Does this imply that the Wronskian tells linear independence only in a particular space that is specific to the solution set, in which the solution set to a second order ODE would be a 2-space?

 

[lurflurf]

The Wronskian is not a conclusive test. That is when the Wronskian is identically zero we draw no conclusion. Often we impose stonger conditionns on the functions (like being analytic or normal solutions of a differential equation) so that the test becomes sufficient.