Question

Compute the Wronskian of any two solutions without solving the equation: dk dx da2 山

Please show all work. I understand how to do the wronskian itself, but confused as to how to find the values for the wronskian from the given diff eq.

Answer 1

As w'/w = -a_1/a_0 Here a_1 = -10x, a_0 = (9 - 8x^2) Thus w'/w = 10x/9 - 8x^2 Integrating both sides integral w'/w dw = integral 10x/9 - 8x^2 dx implies ln w (x) = integral 10x/9 - 8x^2 dx put 9 - 8x^2 = t - 16x dx = dt implies x dx = -dt/16 implies ln w (x) = -1/16 integral dt/t implies ln w (t) = -1/16 ln t + ln c implies ln w (x) = ln t^- 1/16 + ln c implies w (x) = c (9 - 8x^2)^- 1/16