To be linear, $\displaystyle L(\alpha u+\alpha v)=\alpha (L(u)+ L(v)) \ \alpha\in\mathbb{R}$.

How is this equation (see below) linear?

$\displaystyle yu_{xxy}-e^xu_x+3=0$

$\displaystyle yu_{xxy}-e^xu_x+3=0 \ \mbox{and} \ yv_{xxy}-e^xv_x+3=0$

$\displaystyle L(u+v)=(yu_{xxy}-e^xu_x+yv_{xxy}-e^xv_x)+3\Rightarrow y(u_{xxy}+v_{xxy})-e^x(u_x+v_x)+3$

$\displaystyle \neq L(u)+L(v)=y(u_{xxy}+v_{xxy})-e^x(u_x+v_x)+6$