Author Topic: TUT0601 quiz3  (Read 325 times)

xuanzhong

  • Jr. Member
  • **
  • Posts: 12
  • Karma: 1
    • View Profile
TUT0601 quiz3
« on: October 11, 2019, 06:29:27 PM »
Find the Wronskian of two solutions of the given differential equation without solving the equation.
$$
cos(t)y^{\prime\prime}+sin(t)y^\prime-ty=0
$$

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
$$
W[y_1,y_2\ ](t)=cexp(-\int p(t)dt)
$$
$$
=cexp(-\int\frac{sin(t)}/{cos(t)}dt〗)
$$
$$
=cexp(\int\frac{1}{cos(t)}\ d(cos(t)〗)
$$
$$
=ce^{ln|cos(t)|}\
$$
$$
=ccos(t)
$$
« Last Edit: October 11, 2019, 07:02:23 PM by xuanzhong »