Finding prime integrals for a sysyem

Question

Hello This might be an easy question, but I cant find a second prime integral for: $$\frac{dx}{2yz}=-\frac{dy}{xz}=-\frac{dz}{xy}$$ For the second one is easy if we take the first two and get: $$\frac{dx}{2y}=-\frac{dy}{x}$$$$\frac{x^2}{4}+\frac{y^2}{2}=c_1(x,y,z)$$ The $2$ is a making a problem and it doesnt seem anything obvious to me, could you help?

Answer 1

In the same way you can take the last two to get $z^2-y^2=c_2$. After that you get to insert the initial conditions into $$z^2-y^2=c_2=\phi(4c_1)=\phi(x^2+2y^2).$$

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The initial conditions you should get from the complete task description. Along the PDE for $z=u(x,y)$ $$2yuu_x−xuu_y+xy=0\text{ or }2yzp−xzq+xy=0$$ it should contain some curve $z_0(r)=u(x_0(r),y_0(r))$ in the solution surface, for instance by specifying $x_0,y_0,z_0$ directly as functions.

Now $c_1,c_2$ are constants along the characteristic curves. The solution surface $z=u(x,y)$ is made up (in a regular situation) of a one-parameter family of characteristic curves. Thus there has to be a dependence between the parameters that you can write as $Φ(c_1,c_2)=0$ or $c_2=ϕ(c_1)$ or $c_1=ψ(c_2)$ or ... The correct function for the dependence is, at least in the explicit formulations, determined by the initial conditions. This in turn allows to identify the solution of the PDE.