Author Topic: TUT0601 Quiz3  (Read 3870 times)

Joy Zhou

  • Jr. Member
  • **
  • Posts: 5
  • Karma: 0
    • View Profile
TUT0601 Quiz3
« on: October 11, 2019, 09:47:00 PM »
Find the Wronskian of two solutions of the given differential equation without solving the
\cos (t) y^{\prime \prime}+\sin (t) y^{\prime}-t y=0

First, we divide both sides of the equation by $\cos (t):$
y^{\prime \prime}+\tan (t) y^{\prime}-\frac{t}{\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

Noting if we let $p(t)=\tan (t)$ and $q(t)=-\frac{t}{\cos (t)},$ then $p(t)$ is continuous
everywhere except at $\frac{\pi}{2}+k \pi,$ where $k=0,1,2, \ldots$ and $q(t)$ is also continuous
everywhere except at $t=0$.

Therefore, by Abel's Theorem: the Wronskian $W\left[y_{1}, y_{2}\right](t)$ is given by
\begin{aligned} W\left[y_{1}, y_{2}\right](t) &=cexp\left(-\int p(t) d t\right) \\ &=cexp\left(-\int \tan (t) d t\right) \\ &=c e^{\ln |\cos (t)|} \\ &=c\cos (t) \end{aligned}