- #1
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!
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!