What is the purpose of the Wronskian in solving second order linear ODEs?

[member 428835]

Hi pf!

If $y_1$ and $y_2$ are homogenous solutions to a (not necessarily homogenous) second order linear ODE, we define the Wronskian as $$\begin{vmatrix}
y_1 y_2\\
y'_1 y'_2
\end{vmatrix}$$. This derivation seems to stem from the pair of equations involving $y_1$ and $y_2$ satisfying the initial conditions $y(t_0) = y_0$ and $y'(t_0) = y'_0$. But we never satisfy the initial conditions with only $y_1$ and $y_2$ until we also have the particular solution. If everything I've said so far is correct, why is the Wronskian defined in this manner?

Thanks so much!

 

[HallsofIvy]

You are mistaken that y1 and y2 must satisfy some given initial conditions. y1 and y2 can be any solutions to the second order equation. The point of the Wronskian is that we need independent solutions to write the general solutions or solutions to initial value or boundary value equations, and two functions, both satisfying the same second order differential equation, are independent if and only if their Wronskian is non-zero.

 

[member 428835]

awesome. thanks!