Is \rhoutt + EIuxxxx = 0 a Linear or Non-Linear Math Problem?

[squenshl]

I'm trying to see if \rhou_tt + EIu_xxxx = 0 is linear or non-linear where \rho, E and I are constants.

I got L(u+v) = \rho\delta²u²/\deltat² + EI\delta⁴u²/\deltax⁴ + \rho\delta²uv/\deltat² + EI\delta⁴uv/\deltax⁴ = Lu + Lv. Does this mean it's linear or is there more to do.

 

[arildno]

That is enough.

 

[squenshl]

Cheers.
What about this one then.
u_t - \alpha^2\nabla^2u = ru(M -u) where \alpha, r & M are constants.

u_t - \alpha^2\nabla^2u - ru(M -u) = 0
L(u+v+w) = u_t(u+v+w) + \alpha^2\nabla^2u(u+v+w) - ru(M-u)(u+v+w) = u_tt + \alpha^2\nabla^2u² - ru²(M-u) + u_tv + \alpha^2\nabla^2uv - ruv(M-u) + u_tw + \alpha^2\nabla^2uw - ruw(M-u) = Lu + Lv + Lw

 

[squenshl]

Are these equation linear or non-linear?

u_t + (1-u)u_x = 0
u_xx + e^xu_tt = sin(x)
u_xx + u_xy + u_yy + u_x = t²

 

[squenshl]

Someone help please.