Construction of ODE using Wronskian

Question

I have seen some construction of a homogeneous second order linear ODE using the Wronskian $W(t)=W[y_1,y_2](t)$ of the linearly independent twice differentiable functions $y_1$ and $y_2$. The ODE is goven by $$y''-\frac{W'(t)}{W(t)}y'+\frac{W[y_1',y_2'](t)}{W(t)}y=0,$$ It is evident for me that $W(t)=e^{-\int p(t)dt}$ necessarily imply $p(t)=-\frac{W'(t)}{W(t)}$, but I am not getting how $q(t)$ obtained as $\frac{W[y_1',y_2'](t)}{W(t)}$ to get $$y''+p(t)y'+q(t)y=0.$$Also how to find a non-homogeneous second order linear ODE in this way?

Answer 1

The second order homogeneous linear differential equation with given fundamental system $(y_1, y_2)$ can be written as $$ \begin{vmatrix}y & y_1 & y_2 \\ y' & y_1' & y_2' \\ y'' & y_1'' & y_2''\end{vmatrix} = 0 \, . $$ Expanding the determinant along the first column gives $$ \begin{vmatrix} y_1 & y_2 \\ y_1' & y_2' \end{vmatrix} \cdot y'' - \begin{vmatrix} y_1 & y_2 \\ y_1'' & y_2'' \end{vmatrix} \cdot y' + \begin{vmatrix} y_1' & y_2' \\ y_1'' & y_2'' \end{vmatrix} \cdot y = 0 $$ and with $W(t)=W[y_1,y_2](t)$ this is $$ W(t) y'' - W'(t) y' + W[y_1',y_2'](t) y = 0 \, . $$