I am working on a optimal control problem, specifically the design of a hypersonic aircraft nose. When minimizing the drag coefficient, I am led to the following system of differential equations: $$ x_1' = -x_2 $$ $$ x_3' = \frac{-4x_2^3}{1+x_2^2} $$ $$ x_3 = \frac{4x_1x_2^2(3+x_2^2)}{(1+x_2^2)^2} $$ with the boundary/initial conditions $$ x_1(0) = a, \qquad 4x_1(l) = x_3(l), $$ where $l$ is just the endpoint, i.e. $x\in[0,l]$. Is it possible to solve this system analytically, or will I have to resort to a numerical solution?
If it helps, the source of these equations comes from the Hamiltonian $$ H(r,u,p) = \frac{4ru^3}{1+u^2}-pu, $$ with the necessary conditions $$ \frac{\partial H}{\partial u} = 0, \qquad p' = \frac{dp}{dx} = -\frac{\partial H}{\partial r} $$ I just relabeled the variables as $x_1 \equiv r, x_2 \equiv u, x_3 \equiv p$.
$$\begin{cases} \frac{dx_1}{dx} = -x_2 \\ \frac{dx_3}{dx}= \frac{-4x_2^3}{1+x_2^2} \\ x_3 = \frac{4x_1x_2^2(3+x_2^2)}{(1+x_2^2)^2} \end{cases}$$ $$\frac{dx_1}{dx_3}= \frac{1+x_2^2}{4x_2^2}= \frac{1}{4x_2^2}+ \frac{1}{4} \quad\implies\quad \frac{1}{x_2^2} =4\frac{dx_1}{dx_3}-1$$ $$x_3 = \frac{4x_1(3 \frac{1}{x_2^2}+1)}{( \frac{1}{x_2^2}+1)^2}= \frac{4x_1(3 (4\frac{dx_1}{dx_3}-1 )+1)}{( 4\frac{dx_1}{dx_3}-1 +1)^2}= \frac12 x_1\frac{6\frac{dx_1}{dx_3}-1}{(\frac{dx_1}{dx_3})^2}$$
$$x_1\left(\frac{dx_3}{dx_1}\right)^2 -6x_1\frac{dx_3}{dx_1}+2x_3=0$$ Or, on a more usual form, with $x_1=X$ and $x_3=Y$ : $$(Y')^2-6Y'+\frac{2}{X}Y=0$$ This kind of non-linear ODE has no simple general solution.
There is an obvious particular solution : $\quad Y=4X\quad\implies\quad x_3=4x_1$ . As a consequence a particular solution of the problem can be found :
$x_3=4x_1=\frac{4x_1x_2^2(3+x_2^2)}{(1+x_2^2)^2} \quad\implies\quad x_2=\pm 1$
$\frac{dx_1}{dx} = -x_2=\mp 1\quad$ and with condition $x_1(0)=a$ : $$\begin{cases}x_1= \mp x+a \\ x_2:=\pm 1 \\ x_3=\mp 4x+4a \end{cases}$$
This an exact solution of the problem if $\quad 4x_1(l)=x_3(l)=4a\mp 4(l)$ .
If $\quad 4x_1(l)=x_3(l)\neq 4a\mp 4(l) \quad$ I am afraid that there is no closed form solution (made of a finite number of standard functions). Then numerical calculus is required.