How to Solve This 2nd Order ODE in Control Systems?

[ChemEng2]

I'm trying to solve this equation analytically, but I can't even find the auxiliary equation or general solution!

View attachment 1867

Km = 0.5
C*e = 0
K2 = 0.03
K1 = 0.05
x* = 49

 

[Ackbach]

What is $\tau$ and what is $\zeta$? If they do not depend on $t$, then this equation is fairly straight-forward (just assume $C_{m}=e^{kt}$, plug in, and solve for $k$, followed by finding a particular solution of the form $C_{m}=A$.)

 

[Prove It]

I'd also advise substituting the values for your parameters, the equation will be much easier to read at least...