Understanding Transfer Functions in ODEs

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The discussion focuses on deriving a transfer function from an ordinary differential equation (ODE) for an undergraduate design project. The ODE is expressed as A*Y = M*X'' + (B+C)*X' + D*X, with constants A, M, B, C, and D. After applying the Laplace transform, the participant derives the transfer function TF(s) = X(s)/Y(s). The correctness of this derivation is questioned, inviting feedback from others. The exchange emphasizes the importance of validating the transfer function in the context of control systems.
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Hi,

I am undertaking an undergrad design project and want to ensure I'm doing this correctly,

I have derived the ODE as shown below;

A*Y = M*X'' + (B+C)*X' + D*X
Where A,M,B,C,D are constants.

By taking the Laplace transform, I get;
A*Y(s) = M*s^2*X(s) + (B+C)*s*X(s) + D*X(s)

Therefore,
My transfer function becomes;

TF(s) = X(s)/Y(s) = the image of the transfer function below;
http://imageshack.us/a/img37/8820/95533304.jpg


Am I correct here?
 
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Thread 'Proving that convexity implies second order derivative being positive'

There are probably loads of proofs of this online, but I do not want to cheat. Here is my attempt: Convexity says that $$f(\lambda a + (1-\lambda)b) \leq \lambda f(a) + (1-\lambda) f(b)$$ $$f(b + \lambda(a-b)) \leq f(b) + \lambda (f(a) - f(b))$$ We know from the intermediate value theorem that there exists a ##c \in (b,a)## such that $$\frac{f(a) - f(b)}{a-b} = f'(c).$$ Hence $$f(b + \lambda(a-b)) \leq f(b) + \lambda (a - b) f'(c))$$ $$\frac{f(b + \lambda(a-b)) - f(b)}{\lambda(a-b)}...
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