Using Abel's Theorem, find the Wronskian

  • #1
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Using Abel's thrm, find the wronskian between 2 soltions of the second order, linear ODE:
x''+1/sqrt(t^3)x'+t^2x=0
t>0


I think I got the interal of 1/sqrt(t^3) to be 2t/sqrt(t^3) but this is very different to the other examples I've done where a ln is formed to cancel out the e in the formula
W(x1,x2)= Cexp(-intergral 1/sqrt(t^3) dt)
 
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Answers and Replies

  • #2
RUber
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##p(t) = \frac{1}{\sqrt{t^3} }, q(t) = t^2## fits Abel's formula.
So:
##W(t) = C e^{- \int t^{-3/2}dt } ##
In your solution to the integral, I assume you meant that: ## - \int t^{-3/2}\, dt = 2t^{-1/2} ##. This looks to be the best way to answer the problem.
You will often see the natural log occur in practice problems, but there is no rule stating that it will always be there.
A common format looks like ##p(t) = \frac{f'(t)}{f(t)},## so the integral ends up as ##\ln f(t)##, so ##e^{- \ln f(t) } = \frac{1}{f(t)}.## But that is not the format you have here. It looks like your solution is good. Remember to be careful with your negative signs, and try to use Tex when possible to make reading easier.
 

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