306FF
FINAL EXAM
Study Guide
1
Problem #1: Solving an initial value problem for a first order linear system of o.d.e.'s
(a)-[5 pts.]-Rewrite the following system in matrix-vector form: x'  ax by, y'  cx  dy . a,b,c and d
are given constants. Assume that the eigenvalues of the matrix of the given system are real and distinct.
(b)-[5 pts.]-Find a linearly independent pair of vector solutions Y1(t), Y2 (t) of the given system that
 v1 
t
have the form Y (t )  e V , V =   .
v 2 
(c)-[5 pts.]-Use the results of (b) to construct the general solution of the given system.
 x0 
(d)-[5 pts.]-Find the values of the arbitrary constants in the general solution so that Y (0)    .
 y0 
(e)-[5 pts.]-Use information from part (b) to classify the equilibrium point (0, 0) of the given system.
(See sections 3.1, 3.2, 3.3)
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Problem #2: Solving an initial value problem for a first order linear system of o.d.e's
(a)-[3 pts.]-Rewrite the following system in matrix-vector form: x'  ax by, y'  cx  dy .
a,b,c and d are given constants. Assume that the eigenvalues of the matrix of the given system are complex.
(b)-[6 pts.]-Find a linearly independent pair of vector solutions Y1(t), Y2 (t) of the given system that have
 v1 
t
the form Y (t )  e V , V =   .
v 2 
(c)-[5 pts.]-Express the general solution in terms of Y1(t) and Y2 (t) .
(d)-[4 pts.]-Find Re Y1(t) and ImY1(t) .
(e)-[3 pts.]-Express the general solution in terms of Re Y1(t) and ImY1(t) .
(f)-[4 pts.]-Classify the equilibrium point (0, 0) of the given system.
(See sections 3.1, 3.4)
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Problem #3: Solving an initial value problem for a first order linear system of o.d.e's
(a)-[4 pts.]-Re-write the following system in matrix-vector form: x'  ax by, y'  cx  dy .
a,b,c and d are given constants. Assume that the matrix of the given system has exactly one real eigenvalue.
 v1 
t
(b)-[5 pts.]-Find a vector solution Y1(t) of the given system that has the form Y (t )  e V , V =   .
v 2 
(c)-[5 pts.]-Find a vector solution Y2 (t) of the given system that has the form
v 
w 
Y (t )  e t W  tV , V =  1 , W =  1  .
v 2 
w 2 
(d)-[4 pts.]-Express the general solution in terms of Y1(t) and Y2 (t) .
 x0 
(e)-[4 pts.]-Find the values of the arbitrary constants in the general solution so that Y (0)    .
 y0 
(f)-[3 pts.]-Classify the equilibrium point (0, 0) of the given system.
(See sections 3.1, 3.5)
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C. O. Bloom
306B
FINAL EXAM
Study Guide
Problem #4: Solving an initial value problem for a 3-dimensional first order linear system of o.d.e's
(a)-[5 pts.]-Rewrite the following system in matrix vector form:
x '  ax  by  cz, y '  dx  ey  fz, z'  gx  hy  kz
where a, b, c, d, e, f , g, h, k are given constants. Assume that the eigenvalues of the matrix of the given
system are real and distinct.
(b)-[12 pts.]-Find 3 linearly independent pair of vector solutions Y1 (t ), Y2 (t ), Y3 (t ) of the given system
that have the form
 v1 
t
Y (t )  e V , V = v 2 
 v 3 
(c)-[3 pts.]-Use the results of (b) to construct the general solution of the given system.
(d)-[5 pts.]-Find the values of the arbitrary constants in the general solution so that
x0 
Y (0)   y0 
 z 0 
where x0, y0 and z0 are given constants.
(See section 3.8)
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Problem #5:
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 matrix of the linearized system at each equilibrium point.
(c)-[6 pts.]-Find the eigenvalues of the 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.
(See section 5.1 )
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Problem #6: 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 ?
(See section 5.1 )
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2
306B
FINAL EXAM
Study Guide
3
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Problem #7:
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.
(See section 5.3)
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Problem #8:
Laplace Transforms
(a)-[15 Pts.]- Calculate the Laplace transform of the solution y (t ) of the following initial value problem:
ay '' by ' cy  f (t ), y(0)  y0 , y '(0)  v0
where a, b and c are given constants, f (t ) is a given function, and y0 and v0 are given initial values.
(b)-[10 pts.]-If Y(s) is the Laplace transform found in part (a) find y(t )  L1[Y ( s)].
(See Section 6.3. Do problems #5 on page 575 and problems #27, 28, 29 and 31 on page 576.)
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C. O. Bloom