Toronto Math Forum
MAT244--2019F => MAT244--Test & Quizzes => Quiz-3 => Topic started by: xuanzhong on October 11, 2019, 06:29:27 PM
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Find the Wronskian of two solutions of the given differential equation without solving the equation.
$$
cos(t)y^{\prime\prime}+sin(t)y^\prime-ty=0
$$
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 second-order 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
$$
W[y_1,y_2\ ](t)=cexp(-\int p(t)dt)
$$
$$
=cexp(-\int\frac{sin(t)}/{cos(t)}dt〗)
$$
$$
=cexp(\int\frac{1}{cos(t)}\ d(cos(t)〗)
$$
$$
=ce^{ln|cos(t)|}\
$$
$$
=ccos(t)
$$