# Solution of the nonlinear 2nd order differential equation

1. Aug 15, 2008

### younginmoon

1. The problem statement, all variables and given/known data

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.

2. Relevant equations

3. 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.

2. Aug 15, 2008

### 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.

3. Aug 18, 2008