Method of characteristic for second order pde

Question

Can I use the method of characteristic to solve second order pdes? For instance I canconsider the equation $$u_t+u_x=u_{xx}$$

Answer 1

For the equation you specify, consider the change of coordinates

$$ \tau = t \qquad y = x - t $$

we have

$$ \partial_t = \partial_\tau - \partial_y $$

and

$$ \partial_x = \partial_y $$

from the change of variables formula. So

$$ u_t + u_x = u_{xx} \implies u_\tau = u_{yy} $$

In other words, your equation is in fact a variable-transformed heat equation and for this method of characteristics will not work (well).