1. ## u_{xxx}+u_x=0

Find the general solution of u(x,y,z) of

$u_{xxx}+u_x=0$

Can I use this method $m^3+m=0$ for PDE?

Thanks.

$m(m^2+1)=0\Rightarrow m_1=0 \ \ m_2=\mathbf{i} \ \ m_3=-\mathbf{i}$

$u(x,y,z)=f(y,z)e^{0}+g(y,z)e^{x\mathbf{i}}\Rightar row u(x,y,z)=f(y,z)+g(y,z)e^{x\mathbf{i}}$

$u(x,y,z)=f(y,z)+g(y,z)e^{x\mathbf{i}}$
Does it satisfy $\displaystyle u_{xxx}+u_x=0$ ?
Does it satisfy $\displaystyle u_{xxx}+u_x=0$ ?
Yes, $u_x=\mathbf{i}e^{x\mathbf{i}}g(y,z) \ , \ u_{xxx}=-\mathbf{i}e^{x\mathbf{i}}g(y,z)$
$-\mathbf{i}e^{x\mathbf{i}}g(y,z)+\mathbf{i}e^{x\mat hbf{i}}g(y,z)=0$