Write the partial differential equation of cylindrical…
Write the partial differential equation of cylindrical surfaces whose generators are parallel to the vector $(l,m,n)$, and find its general solution.
Explanation of Method
A surface is a cylinder with generators parallel to $(l,m,n)$ exactly when its normal is everywhere perpendicular to that direction. This gives a linear first-order PDE with constant coefficients, solved directly by Lagrange's method.
Step 1 · Derive the PDE
For $z=z(x,y)$, the normal is $\mathbf n=(-p,-q,1)$. The surface is cylindrical with generators parallel to $(l,m,n)$ iff $\mathbf n\cdot(l,m,n)=0$: \[ -lp-mq+n=0 \;\Longrightarrow\; lp+mq=n. \]
Step 2 · Characteristic equations and first integrals
\[ \frac{dx}{l}=\frac{dy}{m}=\frac{dz}{n}. \] Directly: $mx-ly=c_1$ and $ny-mz=c_2$.
Step 3 · General solution
\[ \Phi(mx-ly,\;ny-mz) = 0, \] where $\Phi$ is an arbitrary differentiable function — the family of all cylindrical surfaces with generators parallel to $(l,m,n)$.
Verification
With $P=l$, $Q=m$, $R=n$: for $c_1=mx-ly$, $Pc_{1x}+Qc_{1y}+Rc_{1z}=lm-ml+0=0$. $\checkmark$ For $c_2=ny-mz$, $Pc_{2x}+Qc_{2y}+Rc_{2z}=0+mn-nm=0$. $\checkmark$
Result
The PDE and its general solution are: