Using Abel's Theorem, find the Wronskian

In summary, the conversation discusses using Abel's formula to find the Wronskian between two solutions of a second order, linear ODE. The integral of 1/sqrt(t^3) is found to be 2t/sqrt(t^3), which is different from other examples where a ln is formed to cancel out the e in the formula. However, the solution provided is deemed to be correct. The presence of a natural log in practice problems is not a rule, as seen in this case. The format of p(t) = f'(t)/f(t) is not applicable here, and the solution provided is considered to be accurate.
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NiallBucks
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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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##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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Related to Using Abel's Theorem, find the Wronskian

1. What is Abel's Theorem?

Abel's Theorem, also known as Abel's identity or Abel's formula, is a mathematical theorem that relates the Wronskian of a set of functions to their derivatives. It is named after the Norwegian mathematician Niels Henrik Abel.

2. How is Abel's Theorem used to find the Wronskian?

To use Abel's Theorem to find the Wronskian, we first need to find the derivatives of the given set of functions. Then, we can plug these derivatives into the formula for Abel's Theorem, which is W(f1, f2, ..., fn) = C * exp(-∫f1(x)dx), where C is a constant and f1, f2, ..., fn are the functions in the set. This will give us the value of the Wronskian at any given point.

3. What is the Wronskian and why is it important?

The Wronskian is a mathematical tool that is used to determine the linear dependence or independence of a set of functions. It is important because it helps us to understand the behavior of these functions and their relationship to each other.

4. Can Abel's Theorem be applied to any set of functions?

Yes, Abel's Theorem can be applied to any set of functions as long as they are differentiable. However, it is most commonly used with linearly independent functions.

5. Are there any limitations to using Abel's Theorem to find the Wronskian?

One limitation of using Abel's Theorem is that it can only be applied to functions that are continuous and differentiable on a given interval. Additionally, it may not be the most efficient method for finding the Wronskian in every situation, and other methods may be more suitable.

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