I have been re-learning Maple lately as I am more used to solving things numerically in C/C++ or Fortran so please forgive me if my question is too simple, but...... I have a second-order, autonomous, non-linear ODE (well actually when the operator equation is brought into a cylindrical coordinate system it is non-autonomous) and I keep getting an unevaluated expression with "RootOf" in it. I have tried splitting it into a system of two first-order ODEs and for the autonomous case I was able to obtain a transcendental expression for the second derivative. My question is if anyone has any ideas of how to solve the ode completely in explicit form. Everything in the solution is real-valued too. alpha, Dcj and Dd are all constants. alpha > 0 as are Dcj and Dd but Dcj may be larger or smaller than Dd. The non-autonomous case is the most important.
ode:=diff(zs(r),r$2) = sqrt(1+(diff(zs(r), r))^2)^2 * Dd/alphaa - sqrt(1+(diff(zs(r), r))^2)^3*Dcj/alphaa
ode:= diff(zs(r),r$2)/(1+diff(zs(r),r)^2) + (1/r)*diff(zs(r),r) = (1/alpha)*(Dcj-Dcj*(1+diff(zs(r),r)^2)
ic:= zs(0)=0, D(zs)(0) = constant a
By the way, I can get a series solution of course but I need to solve it symbolically so that I can do more hand calculations with the result....
I know there is a singularity at r=0 for the cylindrical case and haven't addressed that yet, aside from maybe some sort of coordinate transformation.
In all cases this is a semi-infinite domain on r ( 0 <><>