and λ=1,2,...

What's C(x,t) got to do with X(x)?

Actually, the original PDE Eq is here and after the separation of variables, We have upper Eq as ODD.

∂C/∂t=-V*(∂C/∂x)+ D*(∂2C/∂x2)+ E*(∂4C/∂x4) And the initial condition and boundary condition is:

C(x,0)=0

C(L,t)= g(t)

(∂C/∂x)(0,t)=0

(∂2C/∂x2)(0,t)=0

(∂2C/∂x2)(L,t)=0

E>D>V

With V, D, E, lambda constants and E!=0, consider the polynomial

p(t) = t^4-+D/E t^3 - D/E t -lambda/E.

https://www2.math.upenn.edu/~moose/240S2013/slides8-05.pdf