Toronto Math Forum
MAT2442018F => MAT244Tests => Quiz3 => Topic started by: Victor Ivrii on October 12, 2018, 06:07:20 PM

Find the Wronskian of two solutions of the given differential equation without solving the equation.
$$
t^2y''t(t+2)y'+(t+2)y=0.
$$

quiz answer in the attachment

the new following attachment is right. sorry for my previous mistake to the answer

To find the Wronskian of the equation without solving we can apply Abel's Theorem. However, we must first isolate the second derivative term in $t^2y''(t)  t(t+2)y'(t) + (t+2)y(t) = 0$. We can do this by dividing all terms by $t^2$. Doing so yields the equation $$y'(t)  \frac{t+2}{t}y'(t) + \frac{t+2}{t^2} = 0$$ Now we will compute the Wronskian $$W = ce^{\int p(t)dt }$$ where $p(t) = \frac{t+2}{t}$. Aside: $ \int \frac{t+2}{t}dt = t + 2ln(t)$.
Therefore, we get that $$W = ce^{t + 2ln(t)} = ct^2e^t$$

Qianhao, NO SNAPSHOTS. Next time  will delete. SCAN http://forum.math.toronto.edu/index.php?topic=1078.0
(http://forum.math.toronto.edu/index.php?topic=1078.0)
Yunqi, should not post identical solution to the previous!
Nick, escape ln: \ln