My question is an extension of my old question: How to solve the characteristic equations $\frac{dx_1}{x_1} = \frac{dx_2}{x_2} = \frac{dV}{2V}$?
So, follow the suggested solution, I would get
Given the PDE: $$x_1\frac{\partial V}{\partial x_1}+ x_2\frac{\partial V}{\partial x_2} + x_3\frac{\partial V}{\partial x_3} = 2V,$$ where $V = V(x_1,x_2,x_3)$. The Charpit-Lagrange system of characteristic ODEs are: $$\frac{dx_1}{x_1} = \frac{dx_2}{x_2} = \frac{dx_3}{x_3} = \frac{dV}{2V}.$$
1st, $dx_1/x_1 = dV/2V$ leads to $$C_0 = \frac{V}{x_1^2}$$
2nd, $dx_2/x_2 = dV/2V$ leads to $$C_1 = \frac{V}{x_2^2}$$
3rd, $dx_3/x_3 = dV/2V$ leads to $$C_2 = \frac{V}{x_3^2}$$
4th, $dx_1/x_1 = dx_2/x_2$ leads to $$\frac{x_2}{x_1} = \alpha_1.$$
5th, $dx_2/x_2 = dx_3/x_3$ leads to $$\frac{x_3}{x_2} = \alpha_2.$$
6th, $dx_1/x_1 = dx_3/x_3$ leads to $$\frac{x_3}{x_1} = \alpha_3.$$
So the general solution can be written as
$$C_0=F(\alpha_1, \alpha_2)\, \Rightarrow\, \frac{V}{x_1^2} = F(\alpha_1, \alpha_2) \, \Rightarrow \, V(x_1,x_2,x_3) = x_1^2 F(\alpha_1, \alpha_2)$$
I am not sure if my general solution $V$ is correct. My questions really is:
There are many combinations, $C_0, \ldots, C_2$ and $\alpha_1, \ldots, \alpha_3$. How to write down a general solution? or are there many general solutions?
For higher dimension, I am confused about it. Thanks in advanced.