Find general solution. $$ U_{xx} + U_{yy}+ U_{zz} + 2\left(U_{xy}+U_{xz}+U_{yz} \right) = U$$
This is equivalent to
$$ \left(\dfrac{\partial}{\partial x} + \dfrac{\partial}{\partial y}+\dfrac{\partial}{\partial z}\right)^2 U\left(x,y,z \right) = U\left(x,y,z \right)$$
The characteristic surface is given by
$$ \left(\dfrac{\partial}{\partial x} + \dfrac{\partial}{\partial y}+\dfrac{\partial}{\partial z}\right) W\left(x,y,z \right) = 0$$ This is what I could
How to find solution? Thank you!
We straighten the vector field
$$ x \rightarrow x, \, y \rightarrow f=y-x, \, z \rightarrow g=z-x $$
Determinant of the Jacobi matrix $det(J) = 1$
Then $U \left(x,y,z \right) = V \left(x,f,g \right)$ and $ \left(\dfrac{\partial}{\partial x} + \dfrac{\partial}{\partial y}+\dfrac{\partial}{\partial z}\right) \rightarrow \dfrac{\partial}{\partial x} $
and then equation $$ V_{xx} = V$$ has solution $ V = e^x C_1 \left(f, g \right) + e^{-x} C_2 \left(f, g\right) $ and $$U\left(x,y,z \right) = e^x C_1 \left(-x + y, -x + z \right) + e^{-x} C_2 \left(-x + y, -x + z\right)$$