Review, things you should know about/know how to do for exam 2:
I)
Chapter 4: Second-Order Equations (emphasis on inhomogeneous DEs)
A. Section 4.1, 4.3, 4.4
1. See previous review sheet
2. Understand basics in these sections, especially those that are necessary to find solutions to inhomogeneous systems
B. Section 4.5: Inhomogeneous eqns; Method of undetermined coefficients
1. Know the general procedure
a. Find homogeneous solutions (see previous review sheet for how to do this…Section 4.3)
b. Find a (one) particular solution to the inhomogeneous DEs
i.
Make appropriate guess
a) If rhs = exp(kx)∙(poly of degree n)∙trig(mx)
b) Guess yp = exp(kx)∙(Q(x)∙sin(mx)+ P(x)∙sin(mx))
Q(x) = A1,nxn +A1,n-1xn-1+…+A1,1x+A1,0
P(x) = A2,nxn +A2,n-1xn-1+…+A2,1x+A2,0
Especially be able to handle the special cases (which you did on your hws) where k = 0, poly of degree n =
constant, and/or m = 0. (i.e. you have just an exponential, just a trig function, or just a polynomial on the rhs)
c) If a term in your guess (after multiplying it out) is a homogeneous solution, multiply guess by x and use this as
your new guess. If the new guess has a term that is a homogeneous soln, multiply by x again. Repeat until guess
has no term that is in the homogeneous soln.
c. Plug guess into original inhomogeneous DE and find the values of the (undetermined) coefficients that make lhs = rhs/that
make the solution guess an actual solution of the original inhomo DE
d. Combine particular solution and homogeneous solns appropriately to get general solution (yg = C1yh1+ C2yh2+yp)
e. Use ICs to get solution corresponding to IVP (involves using ICs C1 and C2)
2. Know how to perform each part of the procedure by itself if asked
3. Know how to use the “complex method” to find a particular solution…as a reminder
a. Real valued DE ay’’+by’+cy = d∙trig(ωt) =>
complex valued DE az’’+bz’+cz = d∙eiωt
b. Guess a particular solution to the complex inhomogeneous DEs: zp = Aeiωt.
c. Find A to make lhs = rhs in complex method eqn…may include
i.
Multiplying by complex conjugate
ii.
Using the “transfer function”…see section 4.7
d. Find particular solution (let’s call it yp) to real inhomogeneous DEs:
i.
If rhs = d∙cos(ωt), yp = Re(zp)
ii.
If rhs = d∙sin(ωt), yp = Im(zp)
4. Know how/why complex method works…Memorize Euler’s formula (eiθ= cos(θ)+ i sin(θ)) and be able to use it
5. If there is more than one term on rhs, for finding particular solution
a. If rhs = f1+f2+…fn
b. Find inhomogeneous solution for each rhs function…i.e. find yp1 that solves orig lhs = f1, find yp2 that solve original lhs =
f2, etc
c. A particular soln for original inhomogeneous DE is given by yp = yp1+yp2+…ypn
C. Section 4.6: Variation of Parameters
1. Find homogeneous solutions
2. Find a particular solution to the inhomogeneous DEs
a. Make particular solution guess: yp = v1yh1+v2yh2
b. Plug into original inhomogeneous DE to find the functions v1 and v2. This includes:
i.
Setting v1’y1+v2’y2 = 0 (makes yp’ = v1yh1’+v2yh2’)
ii.
Obtaining another eqn/condition for/on v1 and v2 as derived from the original inhomo DE
c. Integrate to find v1 and v2
d. Put these back into original guess for yp for explicit expression for yp
3. Combine particular solution and homogeneous solns appropriately to get general solution (yg = C1yh1+ C2yh2+yp)
4. Use ICs to get solution corresponding to IVP (involves using ICs C 1 and C2)
5. Know how to perform all of the above by themselves if asked for it
6. Note: you can use formulas 6.16 in section 4.6 on the exam, provided you memorize/use them correctly
D. Section 4.7: Forced Harmonic Motion
1. Be familiar with the following terms (what they mean/correspond to)
a. driving frequency, natural frequency
b. Mean frequency, half difference, envelope
c. Resonance
d. Transfer function, gain
e. Transient and steady-state terms
f. Time constant (be able to find)
2. Given a harmonic differential equation, be able to read off the natural frequency, damping constant (note: damping constant ≠
damping coefficient), and driving frequency
3. Be able to solve a harmonic equation (with forcing) using techniques above
4.
For complex method be familiar with the following:
a. Eqn: x’’+2cx’+ω02x = Acos(ωt)
b. Particular soln to inhomogeneous equation looks like: G(ω)Acos(ωt-φ) . (There are also homogeneous solns.)
c. Transfer function: H = Geiφ
d. You do not need to memorize the formulas for the gain, transfer function, and phi but should know what they correspond
to and how to derive the formulas for these quantities if asked or if needed
e. Know how to combine trig functions of slightly different frequencies; includes memorizing appropriate trig formulas and
remembering how to use them
II) Chapter 5: The Laplace Transform
A. Section 5.1: Definition of the Laplace Transform

1.
Know definition(s) of Laplace Transform
T
L{ f }(s)   f (t )e st dt  lim  f (t )e st dt
0
T  0
Be able to use this “fundamental definition of the Laplace Transform” to find the Laplace transform of simple functions
(polynomials, trig, exponentials, poly∙exp, etc., seriously though, make sure you have your integration down including
integration by parts)
3. Be able to use the definition to find Laplace transforms of discontinuous functions
4. Know definitions of piecewise continuous/differentiable
5. Note: The LT table that will be on the test is available on my website…but it is important to know how to get the formulas on
this table as you may be asked to derive one of the formulas in one of the exam problems
B. Section 5.2: Basic Properties of Laplace Transform
1. Know the basic properties of the laplace transform
a. Laplace transform of derivatives
i.
L( y ' )( s)  sL ( y )  y (0)
2.
b.
L( y' ' )( s)  s 2 L( y)  sy (0)  y' (0)
Linearity of Laplace transform L(f  g )( s )  L( f )( s )  L( g )( s )
c.
Exponential shifts: L{e ct f (t )}( s)  L{ f (t )}( s  c)
d.
Derivatives in reverse: L{t n f (t )}( s)  (1) n
ii.
dn
( L{ f (t )}( s))
ds n
2. Know how to use these properties to take old/already known LTs and find new LTs
C. Section 5.3: Inverse Laplace Transform
1. Know basic def of inverse laplace transform
2. Know properties of inverse laplace transform
a. Linearity (see above)
1
1
b. Function shifts (i.e. exponential shifts in reverse): L {F ( s  c)}(t )  e L {F ( s)}(t )
3. Use these properties along with already known inverse laplace transforms to find new inverse laplace transforms
4. Use partial fraction decomposition to find inverse laplace transforms
5. Note: You will be given a limited table of functions and their LTs…you need to know how to convert those LT pairs into
useful inverse LT pairs
D. Section 5.4: Solving DEs with LTs
1. Be able to solve a differential equation using LT. This involves:
a. Using LTs to convert a DE involving y and one or two of its derivatives into an algebraic equation involving only s and
Y=L(y)(s).
b. Appropriately incorporating ICs into step in (a) above
c. Solving for Y
d. Taking inverse LT of both sides to find original y (again good to know partial fraction decomp)
2. Be able to do all above steps individually if asked to
E. Section 5.5: Discontinuous forcing terms
1. Be familiar with Heaviside function, shifted Heaviside function, and interval functions
a. Know their definitions
b. Know what they look like when plotted
c. Know how to find their LTs
2. Basic formula for taking LT of Heaviside∙normal function
a.
b.
3.
Be familiar with formula and how to find:
ct
L{ f (t  c) H (t  c)}  e cs L( f )( s)
L1{e cs F ( s)}(t )  L1{F ( s)}(t  c) H (t  c)
Be able to use both formulas including manipulation of expressions to make them look right so you can actually use the
above formulas (e.g. L(t∙H(t-2)) must be rewritten as L((t-2)∙H(t-2)+L(2∙H(t-2)) before using above formula
Be able to solve DE when forcing function is discontinuous. This involves:
a. Being able to convert any piecewise function into a function involving only shifted Heaviside functions
b. Being able to find the LTs of general piecewise functions (rewritten using shifted Heaviside functions) using already
known LTs (on table)
c. Being able to find inverse LTs of functions including e-cs of some sort (see point II.E.2.b)
F.
d. All other points listed in II.D.1
4. Be able to do all of the above in II.E.3 by themselves if asked
Section 5.6: The delta function
1. Understand what “impulse” physically roughly represents (as well as integral formula)
2. Know basic definitions of delta function given in class:
a. Approx. delta func:   (t )  (1 /  )( H p (t )  H p  (t ))
b.
Delta function about p >= 0 is the function such that:


0
 p (t ) (t )dt   ( p)
3.
4.
Know how to find LT of delta function
Unit impulse response functions
a. Know definition (solution if you replace the rhs of your original diff eq with the delta function centered at 0 with ICs = 0)
b. Know how to find (solve diff eq with new rhs and ICs)
G. Section 5.7: Convolutions
1.
2.
3.
Know and be able to use basic formula: f  g 
t
o
f (u) g (t  u)du
Know f * g  f  g .
Know LT and inverse LT formulas and how to use:
a. L( f * g )  L( f )  L( g )
b.
4.

L1 ( F  G)  L1 ( F ) * L1 (G)
General solution formula involving the unit impulse response function: e * f  ay 0 e'(ay1  by 0 )e
a. Be able to find formula if asked (in a guided way) for the formula
b. Be able to use formula if it is given to you and you have found or are given the unit impulse response function
c. Do not need to memorize for Exam 2.
III) Chapter 6: Numerical Methods
A. Section 6.1: Euler’s Method
1. Be familiar with terms such as
a. Fixed-step, single-step, step size
b. Numerical solution method/numerical solver
c. Truncation error, propagated truncation error, local truncation error
d. Round-off error
2. Know that numerical integration has less error associated with it when
a. You integrate for less time
b. You take smaller step sizes in your numerical solver
c. Your solution is “smoother”
3. Know how to use Euler’s method to obtain a numerical solution as done in class with a table
a. Memorize iterative equation: y k 1  y k  hf (t k , y k )
b.
c.
Know how to use this equation
Know how to perform Euler’s method graphically by using tangent lines on a graph (especially given a direction field in
the t-y plane…only have to worry about Euler’s method for single DEs, not for systems of equations)
IV) Chapter 8: Introduction to Systems
A. Section 8.1: Definitions and examples
1. Know standard terms describing systems including
a. Linear, nonlinear
b. Autonomous, nonautonomous
c. Homogeneous, inhomogeneous
d. Order, dimension, planar
e. Solution (to the system)
2. Be able to check whether or not a set of functions is a solution to a DE system
3. Be able to rewrite higher order DEs as systems of first order equations
4. Be familiar with SIR and predator-prey models (and be able to interpret their results in terms of what they mean physically)
5. Convert between scalar and vector representation of system
B. Section 8.2: Geometric interpretation of solutions
1. Understand
a. What the phase plane/phase space corresponds to
b. What solution curves are in phase plane/phase space and how to interpret them
2. Be able to interpret/draw (in some instances) solutions plotted in:
a. Time plots
b. Phase plane/space plots
c. 3-d plot (for second dimension systems, (y1,y2,t))
d. Composite graph/plot
3. Draw and/or interpret a direction field and use direction field to draw approximate solution curves
C. Section 8.3: Qualitative analysis—not on exam
V) Chapter 9
A. Section 9.1 and 9.2: Overview of the technique/Planar systems
1. Know how to convert between multiple linear equations and one vector equation including a matrix (e.g. y1’=ay1+by2;


y2’=cy1+dy2…convert to y '  Ay )
2. Know how to interpret solutions from vector system in terms of the original scalar unknown functions


3. Know how to solve y '  Ay :

b.
c.
 v1 

 v2 
Guess y  e t 

a.
Find characteristic polynomial (det(A-λI) = 0 or for 2-dimension systems λ2-Tλ+D=0) and solve for the two eigenvalues
Three cases for finding eigenvectors
i.
Case 1: Two real, distinct eigenvalues. For each eigenvalue

a) Use ( A  I )v  0 to derive a relationship between v1 and v2.
b) Use this to pick a suitable eigenvector for each eigenvalue.

ii.
t
 
t

c) Two fundamental solutions given by y1  e 1 v1 ; y 2  e 2 v 2
Case 2: One real, distinct eigenvalues. For this eigenvalue

a) Use ( A  I )v  0 to derive a relationship between v1 and v2.
1) If we get zero matrix times vector v = 0, then all vectors are eigenvectors. Pick two linearly independent

 

vectors as eigenvectors. Two fundamental solutions given by: y1  e t v1 ; y 2  e t v2
2) Otherwise we cannot find two linearly independent eigenvectors, only one. Use relationship between v1 and


v2 to pick one such eigenvector v1 . After that find a generalized eigenvector v2 by solving

 
 


( A  I )v2  v1 . Two fundamental solutions given by: y1  e t v1 ; y 2  e t (tv1  v2 )
iii.
d.
Case 3: Two complex (distinct) eigenvalues. For one of the eigenvalues

a) Use ( A  I )v  0 to derive a relationship between v1 and v2.
b) Use this to pick a suitable eigenvector


c) Use this to construct one complex solution: z  e ( i ) t v1 .
d) Carefully extract the real and imaginary parts of the solution (involves using exponential properties and Euler’s

 

formula) and then the two fundamental solutions are given by: y1  Re( z ); y 2  Im( z )



Write down general solution (if asked): y g  C1 y1  C 2 y 2
e. Use ICs to find C1 and C2 if asked
Be able to perform all parts above by themselves if asked including finding characteristic polynomials, eigenvalues, and
eigenvectors
5. Be somewhat familiar with mixing problems
6. Note all systems here are linear, constant coefficient, homogeneous, autonomous, planar, first order
B. Section 9.3: Phase plane portraits
1. Be able to draw phase plane portraits for types of system seen in Section 9.1/9.2
2. Be able to classify the equilibrium point (0,0) for a given system as
a. Spiral sink (α < 0)/source(α < 0)/or a center(α = 0) (complex eigenvalues λ=α±iβ)
b. Nodal sink (λ1< λ2<0)/source(λ1>λ2>0)/or a saddle point (λ1>0>λ2) (two real eigenvalues)
c. A nodal sink(λ<0)/source(λ>0) (repeated eigenvalue)
d. Stable vs Unstable vs Asymptotically stable
3. Know the implications of an equilibrium pt being classified as one of the above…what does it tell us about where solutions
came from/where solutions are going
4. Be familiar with terms
a. Half-line solutions/exponential solutions
b. Separatrices/Separatrix
5. Be able to draw phase portrait/a bunch of representative solution curves for a given system by obtaining as much information
necessary to do so
a. For real eigenvalues, need eigenvalues, eigenvectors
i.
Draw half-line solutions
ii.
Consider what solution looks like as t goes to ±infinity
b. For complex valued solutions, need eigenvalues only
i.
Use a test point to find the direction of your solution at that test pt
ii.
Use that and whether or not the solution is a spiral sink, source, or center, to make a best estimate as to what the
solution ought to look like
iii.
Use fact that solutions are “concentric spirals/ellipses” to draw other representative solutions
iv.
If you have exact solution, you may consider using this to assist in drawing an accurate solution.
4.