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July 12th, 2016, 11:48 PM   #1
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solving ODE

I need to solve analytically the following second order ODE;

W''(X) - f(X)*W(X)=0

WHERE f(X) is an arbitrary known function.

Could someone help me as soon as possible?

Last edited by skipjack; July 13th, 2016 at 11:15 PM.
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July 13th, 2016, 01:35 AM   #2
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Have you tried to take a look at e.g. Mathworld (link) and see what you can do?
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July 13th, 2016, 02:23 AM   #3
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I have tried, but there no analytical solution is presented for the same ODE. Kindly, could you show me how to obtain a closed form depending on the function f(X)? This my main objective. The solutions illustrated in the website are obtained under some assumptions and simplifications.

Last edited by skipjack; July 13th, 2016 at 11:14 PM.
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July 13th, 2016, 03:03 AM   #4
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In general, no closed form is possible. The simplest case for which a closed form is possible is where f is a constant.
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July 13th, 2016, 09:42 AM   #5
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Quote:
Originally Posted by ammar View Post
I have tried but, there no analytical solution is presented for the same ODE. Kindly, could you show me how to obtain a closed form depending on the function f(X)? This my main objective. The solutions illustrated in the website are obtained under some assumptions and simplifications.
As skipjack said, there is no solution in closed form. In general you can write a generic solution in terms of power series. For more useful series, you need to know the function $\displaystyle f$.

But the generic solution (be warned, there may be many cases - even trivial ones - when this generic solution simply doesn't work and further analyze is needed):

Write the equation in form $\displaystyle y'' = fy$.

Then the general derivative of the right-hand side can be written as

$\displaystyle g_n(x) = D^n\left(f(x)y(x)\right) = \sum_{v=0}^n \binom{n}{v} \left( D^v f(x) \right) \left( D^{n-v} y(x) \right) $

Then the function $\displaystyle y$ can be written in terms of $\displaystyle x$

$\displaystyle y(x) = C_0 + C_1(x - x_0) + \sum_{n = 2}^{\infty}\frac{g_n(x_0)}{n!} (x - x_0)^n$.

But you need to be extremely careful if the series is defined or not, what is the radius of convergence and so on, so this is not actually a solution.
Thanks from manus

Last edited by skipjack; July 13th, 2016 at 11:14 PM.
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July 14th, 2016, 11:02 AM   #6
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Quote:
Originally Posted by fysmat View Post
$\displaystyle y(x) = C_0 + C_1(x - x_0) + \sum_{n = 2}^{\infty}\frac{g_n(x_0)}{n!} (x - x_0)^n$.

But you need to be extremely careful if the series is defined or not, what is the radius of convergence and so on, so this is not actually a solution.
A tiny typo: the formula should read as

$\displaystyle y(x) = C_0 + C_1(x - x_0) + \sum_{n = 2}^{\infty}\frac{g_{n-2}(x_0)}{n!} (x - x_0)^n$.
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July 18th, 2016, 01:41 AM   #7
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First order non linear ODE

I need to solve analytically the following first order non-linear ODE;

F'(X) + F(X)*F(X)=g(X)

WHERE g(X) is an arbitrary function.

Could someone help me as soon as possible?
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July 22nd, 2016, 09:14 AM   #8
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Quote:
Originally Posted by ammar View Post
I need to solve analytically the following first order non-linear ODE;

F'(X) + F(X)*F(X)=g(X)

WHERE g(X) is an arbitrary function.

Could someone help me as soon as possible?
Google? https://en.wikipedia.org/wiki/Riccati_equation
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