Solution of the nonlinear 2nd order differential equation

[younginmoon]

Homework Statement

I'd like to solve the following non-homogeneous second order differential equation and may
I ask smart scholars out there to help me with this?


y"(1-1.5(y')^2)=Cx^n, (^ denotes "to the power of")

where C and n are constants, and the boundary conditions are:
y=0 at x=0,
y'=0 at x=L/2 (L is between 100 and 200).

Thanks.

Homework Equations

The Attempt at a Solution

Indtroducing v=y', the equation becomes
v'(1.0-1.5v^2)=Cx^n
Integration of the above equation provides
(v-0.5v^3)=nCx^(N+1)-const.
Employing v=0 at x=L/2, const=nC(L/2)^(n+1), and the equation becomes
v-0.5v^3=nCx^(n+1)+nC(L/2)^(n+1)
I can't go any further.

 

[cellotim]

I would suggest finding the roots of the cubic equation. It's already in depressed form (x^3 + bx + c = 0), so it should be easy to solve with the cubic formula. Then all you need to do is integrate the three solutions for v. BTW, you made a mistake integrating the RHS.

 

[younginmoon]

Thank you for the reply.
However, the only method I know to solve this kind of depressed cubic equation is the so-called Cardino formula that becomes a monster when you write it down by following the formula, and it's impossible to integrate. Just try to spread out the formula and apply each term, then you'll see the complexity of solution.
Anyway, thanks !