find general solution of:
u_x + x u_y + xy u_z = xyzu
i've tried with this
dx=dy/(x)=dz/(xy)=du/(xyzu) ...but didn't get the correct solution
i appreciate any help...thanks
Picking it up from your first post - the characteristic equations are:
$\displaystyle
\frac{dx}{1} = \frac{dy}{x} = \frac{dz}{xy} = \frac{du}{xyz}
$
We'll pick in pairs
1) $\displaystyle
\frac{dx}{1} = \frac{dy}{x} \;\; \text{so}\;\; x^2 - 2y = c_1
$
2) $\displaystyle
\frac{dy}{x} = \frac{dz}{xy} \;\; \text{so}\;\;y^2 - 2z = c_2$
3) $\displaystyle
\frac{dz}{xy} = \frac{du}{xyzu}\;\; \text{so}\;\; ue^{-z^2/2} = c_3
$
The solution $\displaystyle c_3 = F(c_1,c_2)$ gives $\displaystyle ue^{-z^2/2} = F(x^2 - 2y , y^2 - 2z)$ leading to my solution.