Solving Nonlinear Diff. Equation with Boundary Conditions

[astromanish]

Hi,

Can somebody guide me in getting solution for the following
non-linear differential equation?

u''=K/((a-u)^2) ...K is a constant

Where u is a function of x and the boundary conditions are
u(0)=0
u(L)=0

I need solution in form of 'u' as a function of 'x' with two constants, which are to be determined from the boundary conditions.

Thanks

 

[arildno]

Multiply with u', integrate, and see if you can do something more.

 

[arildno]

To help you along a bit:
$$u'u''=\frac{Ku'}{(a-u)^{2}}\to\frac{1}{2}(u')^{2}=C_{1}+\frac{K}{(a-u)}=\frac{C_{2}-C_{1}u}{a-u},C_{2}=C_{1}a+K$$
Or, we get:
$$u'=\pm\sqrt{\frac{C_{3}-C_{4}u}{a-u}},C_{3}=2C_{2},C_{4}=2C_{1}$$

Rewriting, and integrating, we have:
$$\int\sqrt{\frac{a-u}{C_{3}-C_{4}u}}du=x+C_{5}$$
You may crack the integral at your left-hand side using the substitution:
$$v=\sqrt{\frac{a-u}{C_{3}-C_{4}u}}$$
since we have:
$$u=\frac{a-C_{3}v^{2}}{1-C_{4}v^{2}}$$
Thus, the problem is resolved to partial fractions decomposition, and an implicit formula of the solution function.
You might well encounter some non-trivial conditions placed on K and a when imposing the boundary conditions.

Good luck!