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Math 64 ODEs Sample Exam Questions
1. Find the general solution: y 00 + y = sec x.
2. In the table below, the first column gives the roots of the characteristic equation for an ODE
L[y] = g; the second column gives g. In the space provided, give the form of the particular
solution that you would use to apply the method of undetermined coefficients. You do not
have to solve for the particular solution.
Roots of CE for L[y](x) = g(x)
g(x)
2, −2, 3 − 2i, 3 + 2i
ex sin 2x
1, 1, 2
x3 ex + x2
+i, +i, −i, −i, 1, −3
x2 cos x
Form of yp for method of undetermined coefficients
3. Prove or give a counter example: Let y = φ(x) and y = ψ(x) be linearly independent solutions
of the ODE y 00 + p(x)y 0 + q(x)y = 0, where p and q are continuous on an open interval I.
Suppose that x0 ∈ I is a zero of φ; then ψ cannot have a relative extreme value at x0 . (Hint:
Make a sketch and then think about the Wronskian)
4. Find the first five terms in the power series solution to the initial value problem:
y 00 − x2 y 0 − y = 0,
y(0) = 1,
y 0 (0) = −1.
5. Find the general solution to the Euler equation x2 y 00 + 5xy 0 + 4y = 0.
6. Determine the indicial equation for the ODE 2x2 y 00 − xy 0 + (1 + x)y, and give the first two
terms of each of two linearly independent series solutions expanded about the singular point
x = 0.
7. Find the general solution of the system
1 1
4 1
0
x =
!
x
In the space provided, give an accurate sketch of the phase portrait for this system and classify
the equilibrium point.
1
x2
6
- x1
Classification of equilibrium:
8. Find the general solution of the system
x0 =
−1/2
1
−1 −1/2
!
x
In the space provided, give an accurate sketch of the phase portrait for this system and classify
the equilibrium point.
x2
6
- x1
Classification of equilibrium:
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9. Find the general solution of the system
1 −1
1
3
0
x =
!
x
In the space provided, give an accurate sketch of the phase portrait for this system and classify
the equilibrium point.
x2
6
- x1
Classification of equilibrium:
10. To stimulate your interest in linear systems, consider the following model for the dynamics
of love affairs.
Bill is in love with Hillary. To set the stage, we will denote by C(t) and S(t), respectively,
the state of Hillary’s love for Bill and of Bill’s love for Hillary. We will assume that positive
values of these state variables denote degrees of love, and negative values degrees of hate
(or at least, displeasure). A value of zero signifies indifference. Suppose that the more Bill
loves Hillary the more Hillary distances herself from Bill. But when Bill gets discouraged
and backs off, Hillary begins to find him strangely attractive. Assume that Bill tends to echo
her: he warms up when she loves him and grows cold when she hates him. A model of this
star-crossed romance would be
S 0 = bC, C 0 = −cS,
where, to be consistent with the story, b, c > 0. In all questions that follow, provide a complete
analysis, taking into account all possible values of the parameters, and draw sketches, as
appropriate, to illustrate your answers. Your analysis must be mathematical, and not just
words.
(a) Give an analysis of Bill and Hillary’s love affair, as described by the equations given
above.
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(b) Suppose Bill and Hillary have a romance that would be described by choosing a = 4, b =
−3, c = 8, d = −6. Explain how this romance evolves, and draw a phase portrait in the
Bill and Hillary, love-hate space. Under what initial conditions will their love prosper?
C
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- S
(c) Now consider the forecast for Bill and Hillary’s love life if it is governed by the general
system
S 0 = aS + bC, C 0 = cS + dC
where the parameters a–d may have any sign. A choice of signs specifies the romantic
styles adopted each of the lovers. For example, if a > 0 and b > 0, then Bill is excited by
Hillary’s love for him, and is further turned on to Hillary by his own affectionate feelings
for her. We will refer to this romantic style as the “eager beaver”. If a < 0 and b > 0,
Bill is a “cautious lover”: he does not trust his own feelings, as stimulated by Hillary,
and resists her magnetic attraction in proportion to his own degree of passion. You may
name the other two romantic styles as you please1 . What happens when two identically
cautious lovers get together? “Identically cautions” means that d = a and c = b. Justify
your answer(s).
(d) Can an eager-beaver Bill find enduring love with a cautious-lover Hillary?
11. Equations2 such as x0 = −x + β tanh x arise in statistical mechanics and in models for neural
networks. We will not worry about the scientific details of such models. Show that this
equation undergoes a supercritical pitchfork bifurcation as β is varied. In the space provided,
give a reasonably accurate plot of the critical points as a function of β (the bifurcation
diagram). What is the leading order approximation to x
ˆ=x
ˆ(β) in the limit of large β?
Label the branches of your diagram as stable or unstable as appropriate. Associate with your
graph the appropriate phase-interval analyses to justify your work.
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2
Bonus points for creative names.
Recall that tanh x = (ex − e−x )/(ex + e−x ).
4
x
ˆ
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- β
12. Determine all the equilibrium points, classify them by linear stability analysis, and make a
plausible sketch of the phase flows for the following system of equations:
x0 = y,
y 0 = x − x3 .
Note any unusual trajectories, such as those that connect equilibria. Find a quantity that is
conserved on trajectories.
y
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- x
Conserved quantity:
13. Prove, in two distinctly different ways, that the following system has no periodic solutions in
the phase plane.
x0 = −x + 4y, y 0 = −x − y 3 .
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14. In class and in the text we analyzed a model of competing species. In that model each species
decreased the growth rate of its competitor. In this problem you will analyze a different
situation, that of mutualistic or symbiotic relationships. In this scenario, each species will
help the other species. The simplest model for mutualism is
x0 = x(1 − σ1 x + β1 y),
y 0 = y(2 − σ2 y + β2 x),
where the parameters i , σi and βi , i = 1, 2, are positive.
(a) Provide an interpretation for the parameters of the model.
(b) Determine conditions under which the model has an equilibrium point with both species
present. Give a biological interpretation of this condition.
(c) Give a nullcline analysis, and an accurate depiction of the phase flows, for the situation
where a two-species equilibrium occurs. Provide the linear analysis for the single-species
equilibria; you do not have to linearize the system about the two-species equilibrium.
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