Answer to Question #104803 in Differential Equations for Suraj Singh

Question

Let y1 and y2 be two solutions of the equation a2(x)y'' + a1(x)y' + a0(x)y =0. If W(y1, y2) is the Wronskian of y1 and y2, show that. a2(x)(dw/dx) + a1(x)W = 0

Answer 1

$W=\begin{vmatrix} y_1 & y_2 \\ y_1' & y_2' \end{vmatrix}=y_1y_2'-y_2y_1'$ and $W'=(y_1y_2'-y_2y_1')'=y_1y_2''-y_2y_1''$

$a_2W'+a_1W=a_2(y_1y_2''-y_2y_1'')+a_1(y_1y_2'-y_2y_1')=$

$=y_1(a_2y_2''+a_1y_2')-y_2(a_2y_1''+a_1y_1')=y_1(-a_0y_2)-y_2(-a_0y_1)=0$