Considering the following ODE : find $f(x)$ such that $$\frac{\sigma^{2}}{2}\frac{d^2}{dx^2}f(x)+a(b-x)\frac{d}{dx}f(x)-(\rho+\lambda)f(x)=-\lambda g(x) $$ Where, $a,b,\rho,\lambda,\sigma\in(0,+\infty)$, and $f(x)$ and $g(x)$ are assumed to have enough "good properties" !

Using the Fourier transform, one specific solution could be found, but I am interested in finding a general solution. It would be great if some one could give me some ideas. Thanks for your time and consideration.


closed as too localized by Victor Protsak, Will Jagy, José Figueroa-O'Farrill, Yemon Choi, Harald Hanche-Olsen Sep 12 '10 at 21:46

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  • $\begingroup$ It's linear in $f$ $\endgroup$ – Will Jagy Sep 12 '10 at 19:11
  • $\begingroup$ Hi Will, I am little confused by your comment, it would be great if you could clarify your comment for me $\endgroup$ – Nameless Sep 12 '10 at 19:16
  • $\begingroup$ The difference of two solutions of an inhomogeneous linear equation is a solution of the corresponding homogeneous equation. $\endgroup$ – Victor Protsak Sep 12 '10 at 19:41
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    $\begingroup$ Nameless, given the elementary nature of this question and your comments indicating lack of understanding of basic facts about linear ODEs, I suggest (a) reading a textbook on ODEs; (b) asking your teachers (if you are a student); (c) posting follow-up questions on other sites listed at the FAQ. Voting to close. $\endgroup$ – Victor Protsak Sep 12 '10 at 20:32

Will + Victor noted the general solution is your particular solution plus the general solution of the corresponding homogeneous DE. And, note, that this homogeneous DE does not depend on $g$. According to Maple, the solution of that homogeneous DE is... $$ f_\mathrm{homog} (x) = C_1 \mathrm{KummerM} \left(\frac{\rho + \lambda}{2 a},\frac{1}{2},\frac{a (b - x)^{2}}{\sigma^{2}}\right) + C_2 \mathrm{KummerU} \left(\frac{\rho + \lambda}{2 a},\frac{1}{2},\frac{a (b - x)^{2}}{\sigma^{2}}\right) $$

  • $\begingroup$ Dear Gerald I thank for your comment but it is really sorry to say the solution you gave makes nonsense for me. All I need are the techniques used to solve the problem. Thanks again $\endgroup$ – Nameless Sep 12 '10 at 20:20
  • $\begingroup$ en.wikipedia.org/wiki/Confluent_hypergeometric_function $\endgroup$ – Will Jagy Sep 12 '10 at 21:27

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