I have a system of two 2'nd order nonlinear DE's, although I'm quite stuck with them. These are supposed to be geodesic equations for a cone. I've computed the Christoffel symbols, plugged them into the geodesic equation and got this. So the DE system is:
$$\frac{d^2u}{d\lambda^2}-\frac{u}{2}\frac{dv}{d\lambda}\frac{dv}{d\lambda}=0$$
$$\frac{d^2v}{d\lambda^2}+\frac{1}{u}\frac{du}{d\lambda}\frac{dv}{d\lambda}=0$$
$$\frac{d^2u}{d\lambda^2}-u\frac{du}{d\lambda}\frac{du}{d\lambda}=0$$ $$\frac{\frac{d^2u}{d\lambda^2}}{\frac{du}{d\lambda}}=u\frac{du}{d\lambda}$$ $$\ln\left|\frac{du}{d\lambda}\right|=\frac12 u^2+\text{constant}$$ $$e^{u^2/2}du=c_1d\lambda$$ $$\int e^{u^2/2}du=c_1 \lambda+c_2$$ $$\sqrt{\frac{\pi}{2}}\text{erfi}\left(\frac{u}{\sqrt{2}} \right)=c_1 \lambda+c_2$$ Fonction erfi : https://mathworld.wolfram.com/Erfi.html $$\boxed{u=\sqrt{2}\:\text{erfi}^{-1}\left(\sqrt{\frac{2}{\pi}} (c_1\lambda+c_2\right)}$$
$$\frac{d^2v}{d\lambda^2}+\frac{2}{u}\frac{du}{d\lambda}\frac{dv}{d\lambda}=0$$ $$\frac{\frac{d^2v}{d\lambda^2}}{\frac{dv}{d\lambda}}=-\frac{2}{u}\frac{du}{d\lambda}$$ $$\ln\left|\frac{dv}{d\lambda}\right|=-2\ln|u|+\text{constant}$$ $$\frac{dv}{d\lambda}=c_3\frac{1}{u^2}$$ $$v=c_3\int \frac{d\lambda}{u^2}+c_4$$ $$v=c_3\int \frac{d\lambda}{\left( \sqrt{2}\:\text{erfi}^{-1}\left(\sqrt{\frac{2}{\pi}} (c_1\lambda+c_2\right)\right)^2}+c_4$$ There is no closed form for this integral in terms of a finite number of standard functions.
If this problem is an academic exercise probably there is a typo in the wording of the question.
If this problem comes from applied sciences or technology solve it thanks to numerical methods of calculus.