The problem that I want to solve is the first-order non-linear ODE given by
P'(x)=A*(P(x)+N) / (P(x)+N/2) - G(x)*(P(x)+B) / (P(x)+N/2)
where P(x) is the unknown to be solved in terms of an arbitrarily specified (but non-negative) endpoint value P(0), A is a constant (if it helps with approximations, I am more interested in positive A than negative A), B and N are positive constants (with B < N), and G(x) is an increasing function of x (not necessarily strictly increasing but we can assume that, together with continuity, if it helps) that satisfies G(0) = 0 but is otherwise arbitrary. Does anybody know how to get an analytical (not numerical) solution for this? To be useful it must apply to arbitrary G satisfying the stated conditions. I already know about various limiting cases or special cases that lead to accurate approximations if not exact solutions, but I don't know how to get an accurate analytical approximation for the solution for the general case. Can you help?
Dear L.D., I think that finding an analytical solution could be an hard topic. Try this: multiplying by (P(x)+N/2) , ode can be written as an integral equation like
1/2 P(x)2 + N/2 P(x) = Integral[ (A - G(x))P(x) + AN - G(x)B]
and for this an iterative method like Picard can be applied for an approximate (analytical) solution. Gianluca
Not sure whether it would help you, but one can interchange independent and dependent variables. Namely, let X=X(p) be the new dependent variable, p the new independent variable, related to your original variables by the formulas X=x, p=P.
Finally, let the function H(p) be such that H(p)=G(x). Then your equation becomes
X'(p)=(p+N/2)/(A*(p+N) - H(p)*(p+B) ), and is readily solved by a simple quadrature.
The usefulness of this to your application depends on the assumption that you can trade an arbitrary function G(x) for an arbitrary function H(p).
Thank you for the suggestions Gianluca and Artur. Regarding the integral equation approach, I have already found many different ways (using various integrating factors) to convert the ODE into an integral equation. The problem with each of these is my inability to find a good initial guess. I have found upper and lower bounds for P using this method, but I have not found upper and lower bounds that are close enough together so that bracketing P between these bounds gives a good enough estimate to serve as an initial guess.
Regarding interchanging dependent and independent variables, I have already tried that. The problem is that I don't know whether a function H(P) represents a given function G(x) until after I have solved for x as a function of P or vice-versa. But thank you for your reply.
Here is something else that might make my problem easier to solve. I am considering the x interval [0,L] for some positive L. L is required to satisfy the condition that P(x) (which depends on the non-negative P(0) together with all other parameters in the equation) is non-negative for all x in [0,L]. As an alternative statement, L is an arbitrary positive number but the allowed values of A are restricted by the condition that P(x) is non-negative for all x in [0,L]. What might make the problem easier is that I would be happy just to find out what A must be in order to satisfy P(L)=0. If there is a way to get an estimate of this A without having to solve for P(x) (e.g., a variational method), that would be great. But I haven't found any method that doesn't require a good initial guess for P(x), and I haven't found a good initial guess for P(x).
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