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Note the following review problems DO NOT cover all problem types which may appear on the    
final.


6.3 preliminaries:

1a.)  Suppose $f(t) = t^2$, then $f(t-2) = \underline{\hskip 2in}$

1b.)  Suppose $f(t) = t^2 + 3t + 4$, then $f(t-2) = \underline{\hskip 2in}$

1c.)  Suppose $f(t) = sin(t) + e^{8t}$, then $f(t-2) =  \underline{\hskip 2in}$

2a.)  Suppose $f(t-2) = (t-2)^2$, then $f(t) = \underline{\hskip 2in}$

2b.)  Suppose $f(t-2) = (t-2)^2 + 3(t-2) + 4$, then $f(t) = \underline{\hskip 2in}$

2c.)  Suppose $f(t-2) = sin(t-2) + e^{8(t-2)}$, then $f(t) =  \underline{\hskip 2in}$

3a.)  Suppose $f(t-2) = t^2 + 2t + 5$, then $f(t) = \underline{\hskip 2in}$

3b.)  Suppose $f(t-2) = 3t^2 + 8t + 1$, then $f(t) = \underline{\hskip 2in}$

3c.)  Suppose $f(t) = cos(t) + 4^{8t}$, then $f(t) =  \underline{\hskip 2in} $


Chapter 6:


4.) Find the LaPlace transform of the following:

4a.)  ${\cal L}(u_3(t^2 - 2t + 1)) =  \underline{\hskip 2in} $

4b.)  ${\cal L}(u_4(e^{-8t})) =  \underline{\hskip 2in} $

4c.)  $ {\cal L}(u_2(t^2e^{3t})) =  \underline{\hskip 2in} $




5.)  Find the inverse LaPlace transform of the following:

5a.)  $ {\cal L}^{-1}(e^{-8s}{1 \over s - 3}) =  \underline{\hskip 2in} $

5b.)  ${\cal L}^{-1}(e^{4s}{1 \over s^2 - 3}) =  \underline{\hskip 2in} $

5c.)  ${\cal L}^{-1}(e^{s}{1 \over (s - 3)^2 + 4}) =  \underline{\hskip 2in} $

5d.)  ${\cal L}^{-1}(e^{-s}{5 \over (s - 3)^4}) =  \underline{\hskip 2in} $

5e.)  ${\cal L}^{-1}(e^{s}) =  \underline{\hskip 2in} $


6.)  Use the definition and not the table to find the LaPlace transform of the following:

6a.)  ${\cal L}(t^3) =  \underline{\hskip 2in} $

6a.)  ${\cal L}(cos(t)) =  \underline{\hskip 2in} $




7.)  Find the inverse LaPlace transform of the following.  Leave your answer in terms of a 
convolution integral:

7a.)  ${\cal L}^{-1}({1 \over (s-2)(s^2 + 4)}) =  \underline{\hskip 2in} $

7b.)  ${\cal L}^{-1}({1 \over (s-2)(s^2 - 4s)}) =  \underline{\hskip 2in} $

7c.)  ${\cal L}^{-1}({ 2s \over (s-2)(s^2 - 4)s}) =  \underline{\hskip 2in} $

8.) Find $f*g$

8a.)  $4t*5t^4 =  \underline{\hskip 2in} $

8b.)   $5t^4*4t =  \underline{\hskip 2in} $

8c.)  $sin(t)*e^t =  \underline{\hskip 2in} $





Make sure you can also solve a quick differential equation using the LaPlace transform and use any 
of the formulas on p. 304.

Chapter 3:

9.)  Solve the following initial problems:

9a.)  $y'' + 6y' + 8y = 0, ~y(0) = 0, ~y'(0) = 0 $

9b.)  $y'' + 6y' + 9y = 0, ~y(0) = 0, ~y'(0) = 0 $

9c.)  $y'' + 6y' + 10y = 0, ~y(0) = 0, ~y'(0) = 0$

9d.)  $y'' + 6y' + 8y = cos(t), ~y(0) = 0, ~y'(0) = 0 $

9e.)  $y'' + 6y' + 9y = cos(t), ~y(0) = 0, ~y'(0) = 0 $

9f.)  $y'' + 6y' + 10y = cos(t), ~y(0) = 0, ~y'(0) = 0$

3.8: 1-5, 7, 11, 14, 3.9:  1 - 8

Make sure you understand sections 3.8, 3.9

10.)  Solve the following initial problems:

10a.)  $y' + 3y + 1 = 0, ~y(0) = 0$

10b.)  $   , ~y(0) = 0$

*10c.)  $cos(t)y'- sin(t)y = {1 \over t^2}, ~y(0) = 0$

10d.)  $y' = {3x^2 - 2 \over xy - xy^2}, ~y(0) = 0$

\eject
Chapter 1:

11.)  For each of the following, draw the direction field for the given differntial equation.  
Based on
the direction field, determine the behavious of $y$ as $t \rightarrow \infty$.  If this behaviour 
depends on the initial value of $y$ at $t=o$, describe this dependency.


11a.)  $y' = y$

11b.)  $y' = 1$

11c.)  $y' = y(y + 4)$

Chapter 7:

12.)  Transform the given equation into a system of first order equations:

12a.)  $x''' - 2x'' + 3x' - 4x = t^2$

12b.)  $x'''' - 2x'' + 3x' - 4x = t^2$






Make sure you also study exam 1 and 2 as well as everything else.  Remember the above list is 
INCOMPLETE.


* means optional type problem.  If a problem like 9c appeared on the final, it would be in the 
"choose" section.


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