Write the partial differential equation satisfied by…

Write the partial differential equation satisfied by cylindrical surfaces whose generators are parallel to the vector $(1,1,2)$, and find the general solution.

Step 1 · PDE

From Problem 70 with $l=m=1$, $n=2$: $\boxed{p+q=2}$.

Step 2 · Characteristic equations and first integrals

\[ \frac{dx}{1}=\frac{dy}{1}=\frac{dz}{2}. \] From the first two: $x-y=c_1$. \tag{I} From the first and third: $z-2x=c_2$. \tag{II}

Step 3 · General solution

\[ z = 2x + \Phi(x-y), \] where $\Phi$ is an arbitrary differentiable function.

Verification

Let $u=x-y$: $p=2+\Phi'(u)$, $q=-\Phi'(u)$, so $p+q=2$. $\checkmark$ Geometrically, on every generator $x-y=\text{const}$ and $z-2x=\text{const}$, so $dx=dy$ and $dz=2\,dx$ — the generator direction is $(1,1,2)$. $\checkmark$