Can y1 et and y2 sin t be solutions of a differential equa

Can y1 = et and y2 = sin t be solutions of a differential equation y + p(t)y + q(t)y = 0, assuming that p(t), q(t) are continuous on I = (1, 1)? (Hint: Does the Wronskian of y1, y2 vanish anywhere?)

Solution

Wronskian is, W=y1 y2\'-y1\'y2=e^t cos t- e^t sin t=e^t(cos t-sin t)

So for wronskian to vanish on I we need

cos t=sin t which happens for ,t=pi/4

This value of t lies in I hence these two cannot be solutions of the given ode