2065f09review3

2065f09review3 - Exam III Review Sheet Math 2065 The...

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Exam III Review Sheet Math 2065 The syllabus for Exam III is Sections 4.6, 5.1{5.5, the matrix algebra supplement, and 7.1{7.3. You should review the assigned exercises in these sections. Following is a brief list (not necessarily complete) of terms, skills, and formulas with which you should be familiar. ² If f y 1 ( t ) ; y 2 ( t ) g is a fundamental solution set for the homogeneous equation y 00 + b ( t ) y 0 + c ( t ) y = 0 ; know how to use the method of variation of parameters to flnd a particular solution y p of the non-homogeneous equation y 00 + b ( t ) y 0 + c ( t ) y = f ( t ) : In this method, it is not necessary for f ( t ) to be an exponential polynomial. Variation of Parameters: Find y p in the form y p = u 1 y 1 + u 2 y 2 where u 1 and u 2 are unknown functions whose derivatives satisfy the following two equations: ( / ) u 0 1 y 1 + u 0 2 y 2 = 0 u 0 1 y 0 1 + u 0 2 y 0 2 = f ( t ) : Solve the system ( / ) for u 0 1 and u 0 2 , and then integrate to flnd u 1 and u 2 . ² Know what it means for a function to have a jump discontinuity and to be piecewise contin- uous . ² Know how to piece together solutions on difierent intervals to produce a solution of one of the initial value problems y 0 + ay = f ( t ) ; y ( t 0 ) = y 0 ; or y 00 + ay 0 + by = f ( t ) ; y ( t 0 ) = y 0 ; y 0 ( t 0 ) = y 1 ; where f ( t ) is a piecewise continuous function on an interval containing t 0 . ² Know what the unit step function (also called the Heaviside function ) ( h ( t ¡ c )) and the on-ofi switches ( ´ [ a;b ) ) are: h ( t ¡ c ) = h c ( t ) = ( 0 if 0 t < c; 1 if c t and, ´ [ a;b ) = ( 1 if a t < b; 0 otherwise = h ( t ¡ a ) ¡ h ( t ¡ b ) ; and know how to use these two functions to rewrite a piecewise continuous function in a manner which is convenient for computation of Laplace transforms. ² Know the second translation principle (Theorem 8.2.4): Lf f ( t ¡ c ) h ( t ¡ c ) g = e ¡ cs F ( s ) and how to use it (particulary in the form of Corollary 8.2.5): Lf g ( t ) h ( t ¡ c ) g = e ¡ cs Lf g ( t + c ) g as a tool for calculating the Laplace transform of piecewise continuous functions.
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2065f09review3 - Exam III Review Sheet Math 2065 The...

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