How to solve this linear hyperbolic PDE analytically? Is it possible to solve this equation analytically?
$$
u_t = k u_{xx} + \frac{k}{c} u_{xt}
$$
I attempted to solve it for a finite domain and homogeneous B.C with separation of variables but it got very ugly, with complex eigenvalues. I'm wondering if the equation could be solved with the method of characteristics? Or if there is a coordinate transformation which converts this to an easier PDE?
I am interested in solving the equation for a sine function initial condition. Either infinite or finite domain would be okay, whichever is easier.

I realized that the coordinate transformation $x=x_*$ and $t=t_*-\frac{1}{2c} x_*$ transforms this equation into the damped wave equation, which has well-known solutions which can be obtained through separation of variables.
