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How to solve the PDEs with more than two independent variables whose their most-general solutions are known?

For example the PDE $u_{xxx}+u_{yyy}+u_{zzz}-3u_{xyz}=0$ , according to does this PDE have a name?, the most-general solution is $u(x,y,z)=F_1(y-x,z-x)+F_2\left(y-\left(-\dfrac{1}{2}+\dfrac{i\sqrt{3}}{2}\right)x,z-\left(-\dfrac{1}{2}-\dfrac{i\sqrt{3}}{2}\right)x\right)+F_3\left(y-\left(-\dfrac{1}{2}-\dfrac{i\sqrt{3}}{2}\right)x,z-\left(-\dfrac{1}{2}+\dfrac{i\sqrt{3}}{2}\right)x\right)~.$

If I have the conditions $u(0,y,z)=f(y,z)$ , $u_x(0,y,z)=g(y,z)$ and $u_{xx}(0,y,z)=h(y,z)$ , how can I find $F_1$ , $F_2$ and $F_3$ ?

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