As a second part of my problem I end up with the differential equation looking like: $ \frac{d^2 y}{dx^2} + \frac{1}{x}\frac{dy}{dx} - \frac{a}{x^2}y - \frac{c}{x}y + b x e^{-x^2/p^2}y - d e^{-x^2/p^2}y = 0. $ It is more complex that my previous question. Can someone suggestion a solution method for this?
Special Differential Equation (Continued)
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ordinary-differential-equations
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0After transfering the ODE of the form $p(x)\dfrac{d^2y}{dx^2}+q(x)\dfrac{dy}{dx}+r(x)y=0$ to the ODE of the form $\dfrac{d^2z}{dx^2}+f(x)z=0$ by considering the method in http://mathworld.wolfram.com/Second-OrderOrdinaryDifferentialEquation.html#eqn24, http://eqworld.ipmnet.ru/en/methods/methods-ode/Khorasani2003.pdf claims that $\dfrac{d^2z}{dx^2}+f(x)z=0$ have method to solve generally for general $f(x)$ . But how is the reliability of http://eqworld.ipmnet.ru/en/methods/methods-ode/Khorasani2003.pdf? – 2012-10-14
1 Answers
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The added complication makes closed form solutions even less likely, but you still have $x=0$ as a regular singular point with indicial roots $\pm \sqrt{a}$, and corresponding series solutions.