Toronto Math Forum
MAT2442019F => MAT244Test & Quizzes => Quiz3 => Topic started by: Fenglun Wu on October 11, 2019, 02:00:13 PM

Find the Wronskian of two solutions of the given differential equation without solving the equation.
$$cos(t)y'' + sin(t)y'  ty = 0$$
First, we divide both sides of the equation by $cos(t)$
$$y'' + \frac{sin(t)}{cos(t)}y'  \frac{t}{cos(t)}y = 0$$
Then, we have $p(t) = \frac{sin(t)}{cos(t)} = tan(t)$
Therefore, the Wroskian
$$W[y_1, y_2](t) = c \times exp(\int p(t)dt)
$$
$$= c \times exp(\int tan(t)dt)
= c \times exp(lncos(t))
= c \times cos(t)
$$