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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: 20,370 Thanks: 2007 
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  
Senior Member Joined: Dec 2013 From: some subspace Posts: 212 Thanks: 72 Math Focus: real analysis, vector analysis, numerical analysis, discrete mathematics  Quote:
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 righthand 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^{nv} 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  
Senior Member Joined: Dec 2013 From: some subspace Posts: 212 Thanks: 72 Math Focus: real analysis, vector analysis, numerical analysis, discrete mathematics  Quote:
$\displaystyle y(x) = C_0 + C_1(x  x_0) + \sum_{n = 2}^{\infty}\frac{g_{n2}(x_0)}{n!} (x  x_0)^n$.  
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 nonlinear 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  
Senior Member Joined: Dec 2013 From: some subspace Posts: 212 Thanks: 72 Math Focus: real analysis, vector analysis, numerical analysis, discrete mathematics  Quote:
 

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