I was wondering why dsolve do not show this one solution I obtained different way, for this ode.
Maple gives 6 solutions to the ODE. But 5 of them are signular (have no constant of integration in them). I am looking at the last solution it gives, the one with constant of integration.
But when I solve this ODE, after simplifying it, I get different general solution than the last one Maple shows. I thought they might be equivalent, but I do not see how they could be.
So my question is why Maple did not show the simpler solution also? Here is the code
ode_orginal:=1/3*(-2*x^(5/2)+3*y(x)^(5/3))/x^(3/2)*diff(y(x),x)/y(x)^(5/3)+1/2*(2*x^(5/2)-3*y(x)^(5/3))/x^(5/2)/y(x)^(2/3) = 0;
This looks complicated and non-linear, but when I solve for y'(x) it gives much simpler and linear ODE (Have to convert the ODE to D before solving for diff(y(x),x), since Maple can complain otherwise).
And solving the above gives the much simpler solution
Now when using dsolve on the original complicated lookin ODE it gives
if there is any hope that one of the above 6 solutions will be the same as the simpler solution, it has to be the last one, with the _C1 in it, since all the others do not have _C1 (singular solutions).
But solving for y(x) from the last one does not give the simpler solution.
So it is a new general solution for the original ode and it is correct, since odetest gives zero.
I think in the process of solving for y'(x), some solutions got lost. Even though I asked for allsolutions there?.
But my main question is: Should dsolve have also have given the simpler solution? Since that one also satisfies the original nonlinear complicated ode