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 Differential Equations Ordinary and Partial Differential Equations Math Forum

 July 12th, 2016, 10:48 PM #1 Newbie   Joined: Jul 2016 From: qatar Posts: 3 Thanks: 0 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 10:15 PM. July 13th, 2016, 12:35 AM #2 Senior Member   Joined: Dec 2013 From: some subspace Posts: 212 Thanks: 72 Math Focus: real analysis, vector analysis, numerical analysis, discrete mathematics Have you tried to take a look at e.g. Mathworld (link) and see what you can do? July 13th, 2016, 01:23 AM #3 Newbie   Joined: Jul 2016 From: qatar Posts: 3 Thanks: 0 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 10:14 PM. July 13th, 2016, 02:03 AM #4 Global Moderator   Joined: Dec 2006 Posts: 20,919 Thanks: 2203 In general, no closed form is possible. The simplest case for which a closed form is possible is where f is a constant. July 13th, 2016, 08:42 AM   #5
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Quote:
 Originally Posted by ammar 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.

Last edited by skipjack; July 13th, 2016 at 10:14 PM. July 14th, 2016, 10:02 AM   #6
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Quote:
 Originally Posted by fysmat $\displaystyle y(x) = C_0 + C_1(x - x_0) + \sum_{n = 2}^{\infty}\frac{g_n(x_0)}{n!} (x - x_0)^n$. 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

$\displaystyle y(x) = C_0 + C_1(x - x_0) + \sum_{n = 2}^{\infty}\frac{g_{n-2}(x_0)}{n!} (x - x_0)^n$. July 18th, 2016, 12:41 AM #7 Newbie   Joined: Jul 2016 From: qatar Posts: 3 Thanks: 0 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? July 22nd, 2016, 08:14 AM   #8
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 Originally Posted by ammar 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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