I'm confused as to how to approach this problem.
I've been asked to verify the following equation (I'll explain my notation after a few things):
dW/dt = [a1(t) + b2(t)]*W
I am given that:
W(t) = det[[x1(t), x2(t)][y1(t), y2(t)]]
dx/dt = a1(t)*x + b1(t)*y
dy/dt = a2(t)*x + b2(t)*y
My notation is meant to imply the following:
dW/dt, dx/dt, and dy/dt represent the derivatives of the wronskian, x, and y with respect to t.
a1(t), a2(t), b1(t), and b2(t) are functions of the variable t, the numbers are meant as subscripts.
W(t) follows the general format of the wronskian, which is the determinant of the matrix of solutions. In conjunction with that, it is known that
x = c1*x1(t) + c2*x2(t)
y = c1*y1(t) + c2*y2(t)
is a general solution to a system of two homogeneous differential equations.
So, again, I'm interested in verifying that dW/dt = [a1(t) + b2(t)]*W, which the book claims is a simple calculation. I've tried expanding the wronskian, W, in terms of x1(t), x2(t), y1(t), and y2(t), then multiplying by a1(t) and b2(t) as given in the equation I listed above, but that just gets messy, and doesn't seem to match any form of an answer I get for dW/dt in terms of variables.
Any help is most appreciated.