I'm an engineer and not quite familiar with solving systems of differential equations numerically, but I need to write a (fluid dynamics) program which contains a system of 4 implicit ODE's and an algebraic equation, which are:
$$ C_m \, \dfrac{dC_m}{dr} - \dfrac{C_\theta^2}{r} + C_f \, \dfrac{C^2 \, cos(\beta)}{b \, sin(\phi)} + \frac{1}{\rho} \, \dfrac{dp}{dr} = 0 $$ $$ C_m \, \dfrac{dC_\theta}{dr} + \dfrac{C_m \, C_\theta}{r} + C_f \, \dfrac{C^2 \, sin(\beta)}{b \, sin(\phi)} = 0 $$ $$ \frac{1}{\rho} \, \dfrac{dp}{dr} + \frac{1}{C_m} \, \dfrac{dC_m}{dr} + \frac{1}{r} = 0 $$ $$ \dfrac{dh}{dr} + \frac{1}{C_m} \, \dfrac{dC_m}{dr} + \frac{1}{C_\theta} \, \dfrac{dC_\theta}{dr} = 0 $$ and an algebraic function $$ \rho = f(p, T) \;\;\; \text{(imported from real gas libraries)} $$ I would like to use an explicit Runge-Kutta-Solver with C++, but therefore the equations must be explicit. I don't see how I could transfer these equations to a possible format, without having something like $$ y_1' = f(x,y_1,y_2') $$
I would be pleased if anybody could help me.
Regards Clemens