I am trying to solve the following equation analytically. I think the solution shouldnt be that hard but I'm really rusty on these kind of things.
V'' - k*J*V^-1/2 = 0
I have then said u = V' and therefore u' = u(du/dV)
u(du/dV) - k*J*V^-1/2 = 0
now you can use separation of variables but assuming I did it correctly you get a u in the answer and since u = V' you have to start all over.
The goal of this is to prove J proportional to V^3/2
Any help on how to do this is appreciated, got to turn this in tomorrow!
No, sorry for not making that clear. J is constant. k*J essentially could be considered 1 constant in this context just for the final answer J proportional to V^3/2
The independent variable in this case is x -> differentiating with respect to space
... where a is a constant. Setting we have...
... so that the (1) becomes...
In (3) the variables are separable so that the solution is easily found...
The (4) is a firdt order DE the solution of which leads us to y... are You able to proceed?...