Toronto Math Forum
MAT2442019F => MAT244Test & Quizzes => Quiz3 => Topic started by: xuanzhong on October 11, 2019, 06:29:27 PM

Find the Wronskian of two solutions of the given differential equation without solving the equation.
$$
cos(t)y^{\prime\prime}+sin(t)y^\primety=0
$$
First, we divide both sides of the equation by cos(t):
$$
y^{\prime\prime}+\frac{sin(t)}{cos(t)}y^\prime\frac{1}{cos(t)}y=0
$$
Now the given secondorder differential equation has the form:
$$
L[y]= y^{\prime\prime}+p(t)y^\prime\ q(t)y=0
$$
Then by Abel’s Theorem: the Wronskian W[y1,y2](t) is given by
$$
W[y_1,y_2\ ](t)=cexp(\int p(t)dt)
$$
$$
=cexp(\int\frac{sin(t)}/{cos(t)}dt〗)
$$
$$
=cexp(\int\frac{1}{cos(t)}\ d(cos(t)〗)
$$
$$
=ce^{lncos(t)}\
$$
$$
=ccos(t)
$$