# Toronto Math Forum

## MAT244--2019F => MAT244--Test & Quizzes => Quiz-3 => Topic started by: Fenglun Wu on October 11, 2019, 02:00:13 PM

Title: TUT0602 QUIZ 3
Post 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
$$W[y_1, y_2](t) = c \times exp(-\int p(t)dt)$$
$$= c \times exp(-\int tan(t)dt) = c \times exp(ln|cos(t)|) = c \times cos(t)$$