Math 405 ALL Announcements Spring 2004

Schedule	January	February	March	April
Syllabus,  HomeWork, Text Web Site, Maple, Archive, Math News,	Su Mo Tu We Th Fr Sa 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31	Su Mo Tu We Th Fr Sa 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29	Su Mo Tu We Th Fr Sa 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24  25 26 27 28 29 30 31	Su Mo Tu We Th Fr Sa 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 1 2 3 4 5 6 7 8

• Last3/11/2004  5pm   This Week. Mar 9 & 11 .
• "If you do not know where you are going, it does not matter which way you go. "
  - Lewis Carroll.
  

• Organize study groups (one student should send me a message with copies to each student),
  
• each student should send me a resume with contact info. 
  
• email to me regarding this class must have a subject in the form: 405: Topic, sender(s). 
  
• Example1:  405: Resume, Vollmar
• Example2:  405: Minutes, Anderson, Riley, Vollmar.   N.B.: Minutes of Study groups must be cc'ed to all members
• Study groups
• Before Class:: Anderson, Murphy, Pierson
• Before Class:: Arnett, Kuster, Linder, Weisner
• After Class:::::: Boyd, Feese, Johnson
• After Class:::::: Campbell, Lorenz, Venard
• Tues  2pm :::::: Bullock, Walhite, Walker
  
• Thur 1pm::::::::: Harcourt, Passamaneck, Raper  
  
• TR 2-4pm :::::: Akers, Riley, Vollmar
  
• Fri 11am ::::::::: Blakemore, Blom, Fink, Rechtin
  
• Maple intro
• Issues, resolution, communication
• Chapter 1

• Homework Due Tuesday 1/27
  
• 1.1 1, 5, 11, 15, Maple 19
• 1.2 1a, 5-13
• 1.3 alt odd, 19
• 2.1 1, 13, 21, 23, 27, 29, 35, 37
• 2.2 1, 9, 21, 27, 31
• 2.3 3, 7, 14, 19, 24
• 2.4 1, 7, 13, 17, 23-26, 27, 28, 33
• More Chapter 2 Homework.
  

• Extended discussion of 1.2 #7 and 2.1 #29
  
• Projects:  brief preliminary presentation(1/22 2-3minutes), presentation of results(1/29, 5min), written report (2/5)
  
• I recommend that the projects be worked and written using MAPLE, but you may use any tools of your choice.
• The written projects should be professional technical reports.  The oral presentations and written reprorts should focus on statement of the problem(s), issues, methods, models, results, interpretations, conclusions.  Technical details associated with model solution and analysis should receive brief mention in the oral presentation and be relegated to a detailed technical appendix in the written report.
  
• Project assignment by Study group
• Before Class:: Anderson, Murphy, Pierson........... 1-5            Mixture
  
• Before Class:: Arnett, Linder, Weisner................. 6, 7, 8        Investment
• After Class:::::: Boyd, Feese, Johnson................... 9,10,11,12  Loan Payoffs
• After Class:::::: Campbell, Lorenz, Venard............  8, 12, 13    Retirement
  
• Tues  2pm :::::: Bullock, Walhite, Walker..............  15,16,17    Population
  
• Thur 1pm::::::::: Harcourt, Passamaneck, Raper.....  26,29,30    Vertical motion
  
• TR 2-4pm :::::: Akers, Riley, Vollmar....................  31,32         Baseballs
  
• ???? :::::::::::::::::: Missing.........................................  21 -23        More Balls in motion
  
• ..
• Unassigned but available for credit.......................  27, 28         Millikan's oil drops
  
• Unassigned but available for credit.......................  24               Skydive, also check the following link: http://aci.mta.ca/TheUmbrella/Physics/P3401/Investigations/FreefallJH.html
  
• Others unassigned problems for individual credit: 14, 18, 19, 20, 33
  

• 2.3 Models:  Mixture/Diffusion Models

• 2.4 Existence and Uniqueness in Linear vs. Nonlinear equations,
  

1/29/2004 Thursday

• 2.5 Autonomous Equations
• graphing and interpretation of y ' = f(y)
• roots of f(y)=0 correspond to constant (equilibrium solutions
• solution may be stable or unstable
• sketch motion of y on y axis as indicated by sign of y'
• convacity
  
• Population Models, discussion of rational for variable birth rates due to
  

• Homework Assignment Previous Assignment for Chapter 2,
  
• 2.5 odd 3-17
  
• 2.6 alt odd
  
• 2.7 Use Maple dsolve({IVP},numeric): alt odd
  
• 2.8 odd  1-11, any  of 13- 19 you can do
• Chapter 3.Homework
• Solution of logistic equation by partial fractions
• 2.6 exact equations (subscriptsbelow  indicate partial derivatives)
  
• What is an exact equation:  M dx + N dy = 0, looks like
  
• chain rule applied to  H(x,y) = k where M = H_x and N = H_y
  
• How to test for exactness:   N_x - M_y = 0
• How to find H(x,y): 
  
• ad hoc: integrate M wrt x, N wrt y, Match terms
• systematic H = Int(M,x)  + c(y),  diff the previous result wrt y, set equal to N, solve for c', then c by solving the subsequent simple DE.
  
• If not exact, check for integrating factor I which fits (I N)_x - (I M)_y =  I_x N + I  N_x - I_y M - I M_y = 0
• If I is a function of x, I_y = 0 and  I_x / I = (  N_x - M_y ) / N ,  solve for I if RHS is a function of x
  
• If I is a function of y, I_x = 0 and  I_y / I = (  N_x - M_y ) / -M , solve for I if RHS is a function of y.
• Separable equations are exact and linear equations have the integrating factor discussed above.

2/5/2004 Thursday

• Two student presentations
• Brief intro to Maple Basics
• Check out the Maple New User's Tour in the Maple Help Menu.
  

2/10/2004 Tuesday

1. Please Read Minutes, which discusses both study groups and minutes.
   
2. 2.8 Existence and Uniqueness Theorem for First order IVP's

2/12/2004 Thursday

1. Test 1 over chapters 1 and 2 is postponed to next Thursday.
   
2. Homework from Chapters 1& 2 due before the test next Thursday
   
3. Homework for chapter 3 Prior Chapter 2 assigment,
   
1. 3.1: All odd 1, 3, 5, ...
   
2. 3.2: Alt odd 1,5,9,....
   
3. 3.3: Alt odd 1,5,9, ....
   
4. 3.4: Alt odd 1-19, all odd 21-41
   
5. 3.5: Alt odd 1-13, all odd 15-41
6. More Chapter 3.
   
4. Lecture Topics for today from Sections 3.1, 3.2, 3.4, 3.5. Read all these section and start on the homework
   
1. Linear DE's can be written as:  y" + B(t) y' + C(t) y = g(t) .
1. The DE is homogeneous (on an interval I) if g(t) = 0 (for all t in I).
2. Solutions exists on an interval I for which  B(t), C(t) and g(t)  are continuous.
3. A unique solution exists on the interval when the values of the function and its derivative are specified  at a single point in the interval:  y(t₀) = y₀ and y'(t₀) = y'₀ are specified.
   
2. Characteristic equation for 2HOLDECC's (2^nd order Homongeneous Ordinary Linear Differential Equations with Constant Coefficients).
1. The first order HOLDECC equation y' + r y = 0 has y(t) = e^{_{(-r t)}} as a solution, try this in a 2HOLDECC y" + b y + c y = 0 to produce ( r^₂ + b r + c) e^{_{(-r t)}} = 0; setting the first factor equal to zero is the characteristic equation.
2. Read 3.1 for solution technique and possible solution behavior when there are two different real solutions to the characteristic equation. Example.  ( r^₂ + 4 r + 3) = 0......
   
3. Read 3.4 for solution technique and possible solution behavior when there are two real different complex solutions to the characteristic equation. ( r^₂ + 4 r + 5) = 0 ......  
   
4. Read 3.5 for solution technique and possible solution behavior when there is one repeated real solution to the characteristic equation. ( r^₂ + 4 r + 4) = 0 ...
5. If your group has only two members, you can merge with another 2 or 3 member group if you wish.  It is not necessary to merge if your two member group work well. However you cannot meet as a one member group if the other members do not show up.  It is permissible to meet simultaneously with other groups without merging, and if you do so and report it as such in both sets of minutes you can then get credit as having met. Groups meeting simultaneously should be reported  in the minutes as an action item, and the other group members should be listed as visitors.  Both group minutes must reflect this simultaneous meeting.  I encourage you to arrange a simultaneous meeting with other groups for practice.
6. Study Group meetings must meet for at least 30 minutes and you must do something
   

1. General advice: Life is short, if the rest of your group is not serious and you are, join another group. I will request an evaluation (confidential), by each member of the contribution of other group members at the end of the semester.  This should not effect your study group grade, unless the majority reports that you were seldom there and did little.  You do have a responsibility to your group.
2. Lecture topic: Linear Differential operators:
1. An operator L is a function which acts on other functions and returns a function: L: domain --> range.  A prime example of an operator is the derivative operator D(f)(t) = f '(t).  (Note: f is a function, D(f) is a function, D(f)(t) is the value of D(f) at t.  This is precisely the syntax required by MAPLE.)  Another example is the Equality or Identity operator E(f) = f
   
2. The derivative operator is linear in the sense that D(a f + b g) = a D(f) + b D(g) for constants a and b.  In general, any operator L is linear if  L(a f + b g) = a L(f) + b L(g) .
   
3. For operators L and M, we can define the sum, L + M, and composition, L@M, of operators by the usual  rules for forming  sums and composition of functions (remember, operators themselves are functions):

(L + M)(f)  =  L(f) +  M(f) and


(L@M)(f) = L(M(f)).


• The notation L@M is expressive but clumsy and usually the notation LM is used. L@L is written as L^₂.  Using this notation, the second derivative of y is (D²)(y) or just
  
• We can also form new operators by multiplying by constants , i.e., (kL)(f) = k L(f), or even by multiplying by a function, (B*L)(f)(t) = B(t)*L(f)(t).  This can be very tricky since L M ,  kL and  B*L  and meaning often must be inferred from context.
• This can be cleared up (at some notational expense) by defining a (linear) multiplication operator M_g(f)(t) = g(t)*f(t).  If we regard a constant k as a (constant) function, then (M_k@L)(f) = M_k((L)(f)) = k L(f)  and (M_B@L)(f) = M_B(L(f) = B*L(f). We can the see that all the various multiplications can be viewed as the same type of operator compositions. However, we will avoid this type of pedantic formalism in favor of context. It is also just not the best way to think about it.
  
4. It is easy to show that if L and M are linear operators, then kL, aL+bM, L@M, B*L are all linear operators.
5. Linear differential operators can be used to represent  Linear differential equations:
1. Example:  y '  + 4 y  =  0  can be written  L(y) =  0, with  L =  (D + 4 E) where  D and E are the derivative and Equality operators: L(y) = (D + 4 E)(y) = D(y) + 4 E(y) = y ' + 4 y = 0
2. If Q = ( D² + 5 D + 4 E ), Q(y) = 0 is the differential equation y" + 5 y ' + 4 y = 0.  The characteristic equation approach to solution can be written as
   
1. Q( e^rt ) = ( r² + 5 r + 4 ) e^rt = 0 , requiring r = -4 or r = -1,
   
2. hence  Q(e^-4t ) =  and Q( e^-t ) = 0,  and Q is linear,
   
3. which implies Q( a e^-4t + b e^-t ) = a Q(e^-4t ) + b Q( e^-t ) = 0
   
3. The operator Q can be "factored", like a polynomial, into (D + 4 E)@(D + E) or (D + E)@(D + 4 E). Equality of the three forms is checked by verifying that both factorizations produe the same result when applied to y.  For instance 

(D + 4 E)@(D + E)(y) = (D + 4 E)(y' + y) = y" + y' + 4 y' + 4 y = y" + 5 y' + 4 y = Q(y)

• Some new Maple worksheets have been posted on the web site.
• 405_0201X_Exercises.mws, local, URL Updated 2/18/04. Exercise 21 (direction fields, initial condition analysis). The URL points to the most recent version of this file.
  
• I am trying to get  Maple at a reduced price for sudents in this course. Savings of up to $50. Do not buy it now.  When I tell you to, you will be able to purchase online and download it. 

• Test 1

1. Please be prepared to return to results presentations related to the paper assignments on Thursday. 
   
2. All groups which will not or do not make preentations on Thursday should schedule an out of class time (during your regular group meeting times is my suggestion) to present results to me.  All result presentations must be completed by next Thurday, March 4, and papers written and submitted by Thursday March 11.
   
3. Lecture Topic Differential equations as linear operators (References Sections 3.2-3.5, 4.1, 4.2)
   
1. Linear differential are operators of the form

  L(y) = A_n(t) y⁽ⁿ⁾ + A_n-1(t) y^(n-1) + .. + A₂(t) y'' + A₁(t) y' + A₀(t) y



2. The Existence and Uniqueness Theorem (EUT)  says there is a solution of L(y) = g(t) on any interval I for which all the coefficient functions A_k(t) and g(t) are continuous and the lead coefficient A_n(t) is nonzero. Usually, the emphasis on the continuity of the coeficients and the nonzero behavior of A_n is enforced by dividing by  A_n(t) and requiring the resulting coefficient functions to be continuous. The solution is  uniquely determined by the "state" of the system at any point t₀ in I. The state of the system is the value of the solution function and its derivatives at t₀.  If y(t) is the solution of an nth order differential equation, the values of y, y', y'', .. y^(n-1) are sufficient to determine the system state, since the values of the higher order derivatives can be determined from the DE itself.. 
   
3. Linear differential operators look like polynomials in D with possibly variable coeffiecients.
   
1. If the coefficients are constant, the linear operators act even more like polynomials, factoring (under functional composition) into lower order operators that commute ( F1@F2 =  F2@F1);
   
2. example L = D² + 3D +2E = ( D + 2E )@( D + E ) = ( D + E )@( D + 2E )
1. This commutative property  reveals the VERY IMPORTANT FACT that all the derivatives of solutions of  HOLDECC are also solutions of the same HOLDECC.  Let L be a  LDOCCs, and L(y) = 0, then
   

L(y^(k)) = (L@D^k)(y) = (D^k@L)(y) = D^k(L(y) = D^k(0) = 0:


4. The structure of the null space:  The set of solutions of L(y) = 0
   
1. The linearity of  L implies that the null space of L, N(L) = {y(t): L(y) = 0 }, is a linear subspace of C^_(n)(I), the set of functions on the interval I with continuous n-th derivatives.
2. The EUT implies that L(y) = 0 has n solutions functions B_k(t), k=0..n-1 which satisfy, at a given t₀ in I,
   

B_k^{( j )}(t₀)  =  1 if j = k, = 0 otherwise

3. These function span the null space: given any set of possible state values, the linearity of  L  implies
   
•  y(t)  = s₀ B₀(t) + s₁ B₁(t) + ... + s_n-1 B_n-1(t)  is a solution, and
  
•  y^{( j )}(t₀) = s₀ B₀^{( j )}(t₀) + s₁ B₁^{( j} (t₀) + ... s_j B_j^{( j )}(t₀) + ... + s_n-1 B_n-1^{( j )}(t₀) = s_j 
  
• so this y is the (unique) solution corresponding to the state vector at t₀.
  
4. These functions are linearly independent since
   
•  y(t)  = s₀ B₀(t) + s₁ B₁(t) + ... + s_n-1 B_n-1(t) = 0 for all t in I, means that the value of y and all of its derivatives are zero at t₀ and these values correspond to the values of the s_j's, hence all s_j = 0.
5. Since the B_k(t) span N(L) and they are independent they form a basis for N(L) , which is an n dimmensional linear subspace of C^₍₂₎(I).
5. Solution of linear nonhomogenous equations:
1. If v and w are both solutions to L(y) = g (corresponding to different ICs), then z = w - v is a solution to L(y) = 0: 
   
1. L( z ) = L( w - v ) = L( w ) - L( v ) = g -g = 0
   
2. Another SUPERPOSITION rule:  if L(y₁) = g₁ and L(y₂) = g₂ then
   
1. L( y1 + y2 ) = L( y1 ) + L( y2 ) = g1 + g2
   This allows a DE like y" + 4 y = 5 + t² +  t³e^(-3t)  to be solved independently for groups of terms on the RHS.
   
6. Solutions of repeated factor linear operators:
1. For ANY linear differential operator (specifically including variable coefficients)

L( t y ) = t L( y ) + L_D( y ) where L_D is a new operator made by  formal differentiation of L by D.


2. If L is an LDOCC and L(y) = 0, then LD(y) is a linear combination of y and its derivatives (see 3.2.1 above), all of which satisfy  L(y) = 0. Consequently, LD(y) is in N(L), the null space of L, and
     L( LD(y) ) = 0.
   
3.  Direct calculation then shows that if L(y) = 0 then L²( t y ) = 0:
1. L( L( t y ) ) = L( t L( y ) + L_D( y ) ) = L( t 0 + LD(y) ) = L( L_D(y)) = 0
   

1. Students in this class may now purchase MAPLE directly for $75.  The UofL  COPYIT centers were selling it (with a printed Manual) for $125.  If you have purchased Version 9 recently for the full price, you may be able to send the Manual back (if it is in mint condition) for a $35 refund.  Contact me for the special URL to the Maple sale site.
   
2. Lecture Topic  See Tuesday

Continued from Tuesday

7. If a 2nd order linear operator behaves like an irreducible quadratics polynomial and does not permit linear   factorization over the real numbers it can be analyzed by completing the square:
1. If Q = (D² + 2b D + b² E + w² E) = ( D + b E )² + w² E  and Q( y ) = 0, then

D²( e^(bt) y ) =  e^(bt) ( y" + 2b y' + b²) = e^(bt) (-w² y ) = - w²  e^(bt) y


2. The operator equation D² = -w² E or D² + w² E = 0 is easy solved by observing that we know the two functions cos(wt) and sin(wt) both satisfy y" = -y.  In fact cos(wt) is one of the basic functions at t₀ = 0 cited in 4.2 above having value = 1 at 0 and derivative = 0.  The other basic function is sin(wt)/w which has value = 0 and derivative = 1.
3. If Q(y) = 0 then z = e^(bt) y  is a solution of  z" + w² z = 0,  so z = e^(bt) y = c1 cos(wt) + c2 sin(wt)
   and y = e^(-bt) (c1 cos(wt) + c2 sin(wt)) which provides two solutions of Q(y) = 0.  That these solutions are sufficient will be established later.
8. The trick with  e^(bt) y can be used to solve all three possible differentiable forms.
   

• Monday 3/1 is the last day to drop.  If you wish to discuss your current status see me Monday.
• Check your email  often.  I am havig difficulty posting to the web site.
  

1. Lecture Topic

1. Homework Note prior homework for chapter 3 on 2/12/04.
1. 3.5 38, 39, 40
   
2. 3.6  alt odd 1- 29, 30, 31, 33-36
3. 3.7  alt odd 1- 17, all 21-28, all 29-32
4. 3.8 alt odd 1-29, do as many of the rest as you can (Group project)
   
5. 3.9 alt odd 1-25, do as many of the rest as you can (Group Project)
6. 4.1 alt odd 1-17, all the rest
   
7. 4.2 alt odd 1-37, 39
8. 4.3 alt odd 1- 17, 19, all 20-22
9. 4.4 alt odd 1-17
   
2. Lecture Topic
1. Exercises with Emphasis on Fundamental solutions,
   
1. operator techniques
2. Characteristic function techniques
   
2. testing for independence with the Wronskian determinant,
1. If the Wronskian of a set of functions  is nonzero at any point, functions are independent.
   
3. Using the wronskian matrix to determine constants given initial conditionsWronskians
4. Analysis of solutions after the constants have been determined
5. Essential Property of the Wronskian of solutions of L(y)=0 :  W ' = -A_n-1(t) W
1. For any n solutions of an n-th order LDE, the Wronskian determinant is either never zero (for a fundamental set) or always zero (a dependent set of solutions)
2. Derivatives of Determinants
   
3. Properties of determinants used for evaluation (easily verified for 2x2 matrices)
   
1. a common factor in a row can be factored out
   
2. Value does not change if a multiple of a row is added/subtracted to/from another row
3. identical rows (or columns) force determinant to be zero
6. Annihilators of common functions.  Given y(t) find a Differential Operator which L(y)=0.  Many different operator may work. Most common function can be, and in our case are, annihilated by LDOCCs - linear differential operators with constant coefficients. Our interest is in least order LDOCCs.
1. Constants: y=A, L =  D
2. Linear polynomials y = A + B t,  L = D²
   
3. n-th degree polynomial are annihilated by D⁽ⁿ⁺¹⁾ .
4. Exponentials e^rt by L = ( D - r E )
5.   t e^rt (and (A + B t )e^rt by L = ( D - r E )².
   
6. If P(t) is any nth deg poly,  P(t) e^rt  is annihilated by  L = ( D - r E )ⁿ⁺¹.
7. cos(wt) and sin(wt) are annihilated by L = D² + w² E .
8. e^rt cos(wt) and e^rtsin(wt) are annihilated by L = D² - 2r D + ( r² + w² ) E .
9. If P(t) is any nth deg poly, then P(t) e^rt cos(wt) and P(t) e^rtsin(wt) are
   annihilated by L = ( D² - 2r D + ( r² + w² ) E )ⁿ⁺¹ .

1. Lecture Topic; Undetermined coefficients (3.6, 4.3)

1. Lecture Topic Reduction of order (3.5) and Variation of parameters(3.7, 4.4)
   

1. Lecture Topic

• HTML Symbol worksheet.