How to solve a differential equation containing an integral term?

I'm asking this question on behalf of another person who is not as familiar as me with computers. He has the following problem to solve and unfortunately Mathcad can't solve such type of equations.

He seeks help to find the function $u = u(x) = ?$ for

$$\frac{d^2u(x)}{dx^2} =\left[U(x) + \alpha^2\right]u(x)$$

where $$U(x) = \frac{2}{x}\left(\exp\left(-\frac{x}{\lambda}\right)-\int_0^x 4\pi\left(u(z)\right)^2\,dz\right)$$

The parameters $\lambda$ and $\alpha$ are real.

Is another tool (i.e. Mathematica) able to solve it ? Do you have another suggestion ?

• well you may use $v(x)=\int u(x)^2 dx$ to get a not easy 'third-order non linear ODE' using $u=\sqrt{v'}$ and derivatives (I didn't reverify this...). – Raymond Manzoni Jun 16 '12 at 9:30

In the case at hand, one could write $\displaystyle U'(x)=-\frac1xU(x)+\frac2x\left(-\frac1\lambda\mathrm e^{-x/\lambda}-4\pi u(x)^2\right)$ and $\displaystyle U(x)=\frac{u''(x)}{u(x)}-\alpha^2$ to deduce the third-order differential equation with no integral term $$u'''(x)=\frac{u''(x)}{u(x)}u'(x)-\frac1x\left(\frac{u''(x)}{u(x)}-\alpha^2\right)u(x)+\frac2x\left(-\frac1\lambda\mathrm e^{-x/\lambda}-4\pi u^2(x)\right)u(x).$$ Or, one could transform this into a system of two equations of lower degree, namely, $$xu''(x)=v(x)u(x),\qquad v'(x)=-8\pi u^2(x)+\alpha^2-\frac2\lambda\mathrm e^{-x/\lambda}.$$