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[tex]\begin{array}{cc}\dot{x}_1=x_2\\ \dot{x}_2=x^3_1+u\end{array}[/tex]

with cost functional

[tex]\frac{1}{2}\int^{\infty}_0 x^2_1 +x^2_2+u^2 \ dt.[/tex]

After looking at some examples, I understand that I have to find a value function V that satisfies (is this correct ?? please help)

[tex]x_2 \frac{\partial V}{\partial x_1} +(x^3_1+u)\frac{\partial V}{\partial x_2} + \frac{1}{2}(x^2_1 +x^2_2+u^2)=0.[/tex]

This is a linear pde of order one. The auxiliary system of ODE is

[tex]\frac{dx_1}{x_2}=-\frac{dx_2}{x_1+\sqrt{3}x_2}=\frac{2dV}{x^2_1 +x^2_2 +(x^3_1+x_1+\sqrt{3}x_2)^2}.[/tex]

I think I can solve the first pair of equations because [itex]\frac{dx_1}{x_2}=-\frac{dx_2}{x_1+\sqrt{3}x_2}[/itex] beause it is homogeneous of degree zero.

I have trouble solving the other pair. Any suggestion or computer program is really appreciated.