D-Operator Notation. Where Yo is the value of the integral at the time, t=0.

Cases of Differential Equations.

Case I. Overdamped.

Case II. Critically Damped.

Case III. Underdamped.

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Sections: Diff. Equations | Laplace Transforms | Transfer Functions

Review of Laplace Transforms

Oliver Heaviside created linear differential equations which developed into Laplace Transforms.

The advantages of Laplace Transforms is that:
1. includes the boundary or initial conditions
2. involces simple algebra and solution
3. work is systematized
4. reduced labor because of the table of transforms
5. includes transient ans steady states in one equation.

S-Notations. S-notation, or complex quantity, consists of the a + jw , basically found on the s- or complex-plane. The requirements under this notation is that:
1) a continuous over every finite interval: 0 £ t₁ £ t £ t₂ and 2) it is in exponential order.

£ [ƒ (t)] = § ƒ (t) e ^-st dt = F(s)

where:
£ [ƒ (t)] is the Laplace Integral
s is a complex-quantity, a + jw

Simple Functions Step Function [u(t)] , Decaying Exponential [e^-at], sinusoid [cos wt] and ramp function [f(t) = t].

Step Function [u(t)].

Decaying Exponential [e^-at].

Sinusoid [cos wt].

Ramp function [f(t) = t]. The solution includes integration by parts, where:
u = t; du = dt; dv = e^-st; v = -(e^-at)/s.

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Sections: Diff. Equations | Laplace Transforms | Transfer Functions

Transfer Functions

Transfer Function. The ratio of the transform of the output, Y(s), to the transform of the input, R(s).

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