Author Topic: Q3 TUT 0801  (Read 2142 times)

Victor Ivrii

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Q3 TUT 0801
« on: October 12, 2018, 06:07:20 PM »
Find the Wronskian of two solutions of the given differential equation without solving the equation.
$$t^2y''-t(t+2)y'+(t+2)y=0.$$

Qianhao Lu

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Re: Q3 TUT 0801
« Reply #1 on: October 12, 2018, 06:23:52 PM »

Yunqi(Yuki) Huang

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Re: Q3 TUT 0801
« Reply #2 on: October 12, 2018, 06:28:18 PM »
the new following attachment is right. sorry for my previous mistake to the answer

Nick Callow

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Re: Q3 TUT 0801
« Reply #3 on: October 12, 2018, 06:35:23 PM »
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 $$W = ce^{t + 2ln(t)} = ct^2e^t$$