Toronto Math Forum
MAT2442018S => MAT244Tests => Quiz3 => Topic started by: Victor Ivrii on February 10, 2018, 05:16:49 PM

Find the Wronskian of two solutions of the given differential equation without solving the equation.
$$
\cos (t)y'' + \sin (t)y'  ty = 0.
$$

Divide everything by $cos(t)$ to get $y''$ by itself.
$$y'' + {sin(t)\over cos(t)}y'  {t\over cos(t)}y = 0$$
Now that it is in the proper form, we can use Abel's theorem of $W = ce^{\intp(t)dt}$ where c is a constant and $p(t)$ is $sin(t)\over cos(t)$ in this case. Now we solve the integral:
$$ce^{\int{sin(t)\over cos(t)}dt}$$
Using the substitution $u=cos(t)$ and $du=sin(t)dt$ we get
$$ce^{\int{1\over u}du}=ce^{ln(u)+C}=ce^{ln(cos(t)+C}=ce^Ccos(t)$$
But $ce^Ce^2$ is just some constant, so we can subsume it into just $c$. Simplifying this, we get that the Wronskian is:
$$W = {c(cos(t))}$$

First, we divide both sides of the equation by $cos(t)$:
$$y''+tan(t)y'{t\over cos(t)}y=0$$
Now the given secondorder differential equation has the form:
$$L[y]=y''+p(t)y'+q(t)y=0$$
Noting if we let $p(t)=tan(t)$ and $q(t)={t\over cos(t)}$, then $p(t)$ is continuous everywhere except at ${\pi\over 2}+k\pi$, where $k=0,1,2,\dots$ and $q(t)$ is also continuous everywhere except at $t=0$. $\\$
Therefore, by Abel's Theorem: the Wronskian $W[y_1,y_2](t)$ is given by
$$\begin{align}W[y_1,y_2](t)&= cexp(\int{p(t)dt})\\&=cexp(\int{tan(t)dt})\\&=ce^{lncos(t)}\\&=ccos(t)\end{align}$$

use \cos \sin etc