306D
FINAL EXAM
Study Guide
1
Do any 5 problems. You may do one, two or three additional problems for extra credit.
Problem #1:
(a)-[15 pts.]-Use Euler’s method with step size t  # to approximate the solution of the following initial
value problem at t  t1 , t2 , t3 ,..., tn where tk 1  tk  t (k  0,1, 2,..., n  1) :
x '  f ( x, y), y '  g ( x, y), x(t0 )  x0 , y(t0 )  y0 .
(b)-[10 pts.]- Plot the initial point ( x0 , y0 ) and the approximate values ( x1 , y1 ),( x2 , y2 ),( x3 , y3 ),( x4 , y4 ).
(See “Systems and Euler’s Method “, section 6.1, p. 309. See table 4 on p. 313)
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Problem #2: Solving an initial value problem for a second order linear differential equation
(a)-[5 pts.]-Rewrite the initial value problem ay  by  cy  0, y(0)  y0 , y(0)  v0 as an equivalent
initial value problem for a 2-dimmensional linear first order system in matrix-vector form. a, b, c, y0 and v0 are
given constants.
(b)-[10 pts.]-Find a linearly independent pair of vector solutions Y1(t), Y2 (t) of the first order system
v 
of the form Y (t )  e tV , V=  1  .
 v2 
(c)-[5 pts.]-Use the results of (b) to find the solution of the first order system that satisfies the
vector initial condition found in part (a).
(d)-[5 pts.]-Use the result of part (c) to find y (t ) and y(t ).
(See sections 9.1, 9.2)
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Problem #3: Solving an initial value problem for a linear second order linear differential equation
(a)-[5 pts.]-Rewrite the initial value problem ay  by  cy  0, y(0)  y0 , y(0)  v0 as an equivalent
initial value problem for a 2-dimensional linear first order system in matrix-vector form. Here a, b, c, y0 and v0
are given constants.
v 
(b)-[5 pts.]-Find a vector solution Y1 (t ) of the given system of the form Y (t )  e tV , V=  1  .
 v2 
(c)-[5 pts.]- Find a vector solution Y1 (t ) of the given system of the form
v 
w 
Y (t )  e t (tV  W ), V =  1  , W   1  .
 v2 
 w2 
(d)-[5 pts.]-Use the results of (b) and (c) to find the solution of the first order system that satisfies the
vector initial condition found in part (a).
(e)-[5 pts.]-Use the result of part (d) to find y (t ) and y(t ).
(See sections 9.1, 9.2. See section 9.4, especially Ex. 4.5 on p. 487. Read pp. 401-402 in sect. 8.1.
Do prob. 11 on p. 403.)
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Problem #4: Solving an initial value problem for a third order linear differential equation
(a)-[5 pts.]- Rewrite the initial value problem ay  by  cy  dy  0, y(0)  y0 , y(0)  y1 , y(0)  y2
as an equivalent initial value problem for a 3-dimensional linear system in matrix-vector form.
Here a, b, c, d , y0 , y1 , y2 are given constants.
(b)-[3 pts.]- Find the eigenvalues of the matrix for the system found in part (a).
2
Hint: The characteristic equation can be written in the form (  A)(  B  C )  0.
(c)-[9 pts.]-Find 3 linearly independent pair of vector solutions Y1 (t ), Y2 (t ), Y3 (t ) of the
system found in part (a) the form
 v1 
t
Y (t )  e V , V=  v 2  .
 v3 
(d)-[5 pts.]- Find the values of the arbitrary constants in the general solution so that the
vector initial condition found in part (a) is satisfied.
(e)-[3 pts.]- Use the result of part (c) to find y (t ), y(t ), y(t ).
(See section 9.4, especially Theorem 2. See Ex. 4.5 on p. 487. Do probs. #23, 25, 29, 31 on p. 491.
Read pp. 401-402 in sect. 8.1. Do prob. 15 on p. 403.)
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C. O. Bloom
306D
FINAL EXAM
Study Guide
2
Problem #5:
 a11

a
Consider the 4-dimensional linear system x '  Ax where A   21
 a31

 a41
a12
a13
a22
a32
a42
a23
a33
a43
a14 

a24 
.
a34 

a44 
Let
 v13 
 v11 
 v12 
 v14 
 
 
 
 
3t  v 23 
1t  v 21 
2t  v 22 
4t  v 24 
x1 = e
,x =e
,x =e
,x = e
 v33  4
 v31  2
 v32  3
 v34 
 
 
 
 
 v 41 
 v42 
 v44 
 v 43 
be four linearly independent solutions of the given system.
(a)-[10 pts.]- Verify by direct substitution that x3 is a solution.
(b)-[10 pts.]- Use the given exponential solutions to find the solution of x '  Ax that satisfies the initial
condition
 x10 
 0
x
x(0) =  20  .
 x3 
 0 
 x4 
(c)-[5 pts.]- Discuss the behavior of the solutions of x '  Ax as t  .
(See section 9.4, especially Theorem 2. See Ex. 4.4 on p. 487. Do probs. #43, 44, 51, 52 on p. 492.
Read pp. 401-402 in sect. 8.1. See probs. 5 and 6 on p. 402. See Ex. 4.17 on p. 367.
Do probs. 40 on p. 384.)
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Problem #6:
Linearizing a non-linear first order system at its equilibrium points.
(a)-[3 pts.]- Locate the equilibrium points of a given non-linear system of the form x'  f (x, y), y '  g(x, y) .
Assume that f (x, y) and g(x, y) are quadratic functions of x and y .
(b)-[3 pts.]-Find the Jacobian matrix of the linearized system at each equilibrium point.
(c)-[6 pts.]-Find the eigenvalues of the Jacobian matrix of the linearized system at each equilibrium point.
(d)-[3 pts.]-Use the results of part (c) to classify each of the equilibrium points.
(e)-[3 pts.]-Sketch the direction field of the given system along the positive y-axis, and along the
negative y-axis.
(f)-[3 pts.]-Sketch the direction field of the given system along the positive x-axis, and along the
negative x-axis.
(g)-[4 pts.]-Sketch the direction field of the given system along each solution plotted on the graph below.
(Read section 5.1, especially “Characterization of Equilibrium Points”. See Ex. 1.18 on p. 550.
Do probs. # 1, 3, 5 on p. 556. See answers to probs. 9, 11 13, 15, 17 on A-54 & A-55.)
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Problem #7:
Classifying the equilibrium points of a non-linear first order system
The phase portrait of two non-linear systems of the form x'  f (x, y), y '  g(x, y) are sketched in
Figures 1 and 2 below.
(a)-[6 pts.]- Where are the equilibrium points of the given system located ?
(b)-[7 pts.]-What kind of equilibrium points does the given system have ?
(c)-[12 pts.]-Discuss the eigenvalues of the linearized system at each equilibrium point.
Are they real or complex ? What are their algebraic signs if they are real?
What can you say about the real parts of any complex eigenvalue ?
(Read section 5.1, especially “Characterization of Equilibrium Points”. See Ex. 1.18 on p. 550.
Do probs. # 1, 3, 5 on p. 556. See answers to probs. 9, 11, 13, 15, 17 on A-54 & A-55.)
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Problem #8:
Hamiltonian Systems
(a)-[5 pts.]- Verify that the given non-linear system x'  f (x, y), y '  g(x, y) is Hamiltonian.
(b)-[7 pts.]- Find the Hamiltonian of the given system.
(c)-[8 pts.]- Locate and classify the equilibrium points of the given system.
(d)-[5 pts.]- Which of the phase portraits sketched below is the phase portrait of the given system?
Briefly explain your answer.
(Read “Hamiltonian Systems “ on pp. 600-605 of section 10.6. See Ex. 6.17 on p. 602.
Do probs. # 7, 9, 13, 23, 27, 29, 30 on pp.608-609.)
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C. O. Bloom