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Unformatted text preview: Differential Equations Review Differential Equations like the following are Linear and Time Invariant: a n d n y t dt n a n − 1 d n − 1 y t dt n − 1 ... a y t = b m d m f t dt m ... b 1 df t dt b f t  coefficients are constants ⇒ TI No nonlinear terms ⇒ Linear Can always ensure that a n = 1 f(t) y(t) LTI System These overheads were originally developed by Mark Fowler at Binghamton University, State University of New York. In the following we review the basics of solving Linear, Constant Coefficient Differential Equations under the Homogeneous Condition. “Homogeneous” means the “forcing function” is zero. Solutions are called “complementary functions”. (In the general case the full solution is the sum of the homogeneous solution and the particular solution found using either the method of undetermined coefficients or by variation of parameters method.) That means we are finding the “zeroinput response” that occurs due to the effect of the initial conditions. ⇒ Write D.E. like this: D n a n − 1 D n − 1 ... a 1 D a = Δ Q D y t = b m D m ... b 1 D b = Δ P D f t Diff . Eq . ⇒ Q D y t = P D f t m is the highestorder derivative on the “input” side n is the highestorder derivative on the “output” side We will assume: m ≤ n d k y t dt k ≡ D k y t Use “operational notation”: Due to linearity: Total Response = ZeroInput Response + ZeroState Response ZI Response : found assuming the input f ( t ) = 0 but with given IC’s ZS Response : found assuming IC’s = 0 but with given f ( t ) applied ⇒ D . E .: Q D y zi t = ⇒ D n a n − 1 D n − 1 ... a 1 D a y zi t = ∀ t (▲) Consider y t = ce λt c and λ are possibly complex numbers “linear combination” of y zi ( t ) & its derivatives must be = 0 Can we find c and λ such that y 0 ( t ) qualifies as a homogeneous solution?...
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 Fall '10
 DRTERRENCETODD
 Complex number, dt dt, characteristic roots, characteristic equation, DT DT DT, Differential Equations Review

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