2065f09review3

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

This preview shows pages 1–2. Sign up to view the full content.

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.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online