Toronto Math Forum
MAT2442019F => MAT244Test & Quizzes => Quiz3 => Topic started by: christine on October 11, 2019, 02:00:10 PM

Question: $cos(t)y''+sin(t)y'ty=0$
Find the Wronskian of two solutions of the given differential equation without solving the equation.
Solution:
Divide both sides by $cos(t)$ and we get: $y''+ tan(t)y'\frac{t}{cos(t)}y=0$
By Abel's theorem, we have: $W(y_1, y_2)(t)=ce^{\int{p(t)}dt}$
$W(y_1, y_2)(t)=ce^{\int{tan(t)}dt}=ce^{(lncos(t))}$
$W(y_1, y_2)(t)=ce^{lncos(t)}=ccos(t)$
Hence, the Wronskian of any pair of solutions of the given equation is $W(y_1, y_2)(t)=ccos(t)$