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

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The discussion focuses on solving a second-order ordinary differential equation (ODE) in control systems, emphasizing the need to find the auxiliary equation and general solution. Key parameters include Km = 0.5, K2 = 0.03, K1 = 0.05, and x* = 49, with a query about the values of τ and ζ. It is suggested that if these values do not depend on time, the problem simplifies significantly by assuming a solution form of Cm = e^(kt) and solving for k. Additionally, substituting the parameter values into the equation is recommended to enhance clarity. The overall goal is to analytically derive the solution to the ODE.
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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
 

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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 $C_{m}=A$.)
 
I'd also advise substituting the values for your parameters, the equation will be much easier to read at least...
 

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