I am trying to solve a system of 1st order ODE's. $m(r)$ and $P(r)$ are real functions of $r$ and should be positive. $a$ and $b$ are just constants.
$$\frac{d m(r)}{dr}=\pi(\frac{P(r)}{a}+b)r^2\\$$
$$\frac{d P(r)}{dr}=-\frac{P(r)}{r-m(r)}\\$$
I could transform the system into one 2nd order ODE. Apparently Mathematica cannot solve this "simple" problem. Could you please advise me to literature, how to learn to solve these systems? Numerically it isn't a problem to solve them with appropriate boundary conditions.
The question I am most interested about is: How can we know, if there are "neat" closed-form solutions to this or similar systems? Is it correct to speak about transcendental ODE's? Can God with infinite intelligence write down functions $m(r)$ and $P(r)$, which solve the above system? I hope you won't attack me, for not having tried harder myself.
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Regarding Mathematica's capability of solving these simple systems, I just post one exact solution of a different system, which Mathematica 7 apparently cannot solve. The solution was obtained by starting from it and then deriving the differential system, which it obeys. Literature recommendations on symbolic computation approaches to solve such systems are very welcome.
$\frac{dm(r)}{dr}=\frac{2r(-1+ar^3-r^4)(10r^2+6a^3r^3-6r^6+a^2(3-18r^4)+6ar(-2+3r^4))}{(1-3a^2r^2+6ar^3-3r^4)^2P(r)}$
$\frac{dP(r)}{dr}=\frac{-P(r)}{r-m(r)}$
with a solution:
$m(r)=\frac{-a+4a^2r^3-8ar^4+4r^5}{-1+3a^2r^2-6ar^3+3r^4}$
$P(r)=r^3+\frac{1}{r-a}$