EECE 301 Differential Equations Review Notes

EECE 301 Differential Equations Review Notes - EECE 301...

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EECE 301 Signals & Systems Prof. Mark Fowler Discussion #3a • Review of Differential Equations
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Differential Equations Review Differential Equations like this are Linear and Time Invariant: ) ( ) ( ... ) ( ) ( ... ) ( ) ( 0 1 0 1 1 1 t f b dt t df b dt t f d b t y a dt t y d a dt t y d a m m m n n n n n n + + + = + + + -coefficients are constants TI -No nonlinear terms Linear . , ) ( ) ( ), ( , ) ( ) ( ), ( etc dt t y d dt t y d t y dt t y d dt t y d t f p p k k n p p k k n Examples of Nonlinear Terms:
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In the following we will BRIEFLY review the basics of solving Linear, Constant Coefficient Differential Equations under the Homogeneous Condition “Homogeneous” means the “forcing function” is zero That means we are finding the “zero-input response” that occurs due to the effect of the initial coniditions. Write D.E. like this: ( ) ( ) ) ( ... ) ( ... ) ( 0 1 ) ( 0 1 1 1 t f b D b D b t y a D a D a D D P m m D Q n n n 4 4 43 4 4 42 1 4 4 4 4 4 4 4 4 1 Δ Δ = = + + + = + + + + . . Eq Diff ) ( ) ( ) ( ) ( t f D P t y D Q = m is the highest-order derivative on the “input” side n is the highest-order derivative on the “output” side We will assume: m n ) ( ) ( t y D dt t y d k k k Use “operational notation”:
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Due to linearity: Total Response = Zero-Input Response + Zero-State Response Z-I Response : found assuming the input f ( t ) = 0 but with given IC’s Z-S Response : found assuming IC’s = 0 but with given f ( t ) applied () 0 0 ) ( ... 0 ) ( ) ( : . . 0 1 1 1 > = + + + + = t t y a D a D a D t y D Q E D zi n n n zi ( ) numbers complex possibly are and ) ( Consider 0 λ c ce t y t = “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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This note was uploaded on 05/07/2008 for the course EECE 301 taught by Professor Fowler during the Spring '08 term at Binghamton.

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EECE 301 Differential Equations Review Notes - EECE 301...

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