306T&Y
TEST #2 - Study Guide
Problem #1:
A Predator Prey Model
See section 2.1 in Text #1
(a)-[6 pts.]- A population of wolves and mice exclusively cohabit the same isolated island.
Translate the following statements into a 1st order system of differential equations:
(i) In the absence of the wolves the mouse population would increase exponentially.
(ii) In the absence of the mice, the wolf population would decrease exponentially.
(iii) Interactions between wolves and mice are favorable to the wolves; the rate of growth of
the wolves is increased by an amount jointly proportional to the wolf and mouse population.
Interactions between wolves and mice are unfavorable for mice; the rate of growth of the
mouse population is decreased by an amount jointly proportional to the wolf and mouse
population.
(b)-[2 pts.]- Sketch the phase portrait for the predator-prey system you obtained in part (a),
where W(t) stands for the Wolf population at time t  0, and M(t) stands for the mouse population at time t  0.
Problem# 1 (continued)
A Competing Species Model
See section 2.2 in Text #1.
(c)-[6 pts.]- A population of sheep and cows exclusively cohabit the same isolated island.
Translate the following statements into a 1st order system of differential equations:
(i) In the absence of the cows the sheep population would increase logistically.
(ii) In the absence of the sheep the cow population would increase logistically.
(iii) Since the cows and sheep feed on the same grassland the interaction between
cows and sheep is unfavorable to both species. The rate of growth of the cow
population is decreased by an amount jointly proportional to the cow and sheep
population. The rate of growth of the sheep population is decreased by an amount
jointly proportional to the cow and sheep population.
(d)-[2 pts.]- For given initial values sketch the solution [S(t), C(t)].
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Problem # 2:
Analytic Methods for Special Systems
An initial value problem for a non-linear, 1st order system of o.d.e.'s
See section 2.3 in Text #1
(a)-[15 pts]- Calculate the exact solution [x(t), y(t)] of the following initial value problem:
x' = f(x), y' = a(x)y + g(x), x(0) = x0, y(0) = y0.
(b)-[5 pts]- Sketch x(t), y(t) and [x(t), y(t)].
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Problem # 3:
Analytic Methods for Special Systems
An initial value problem for a non-linear, 1st order system of o.d.e.'s
See section 2.3 in Text #1
(a)-[15 pts.]- Solve the following initial value problem: x' = f(y) x, y' = g(y), x(0) = x0, y(0) = y0 .
(b)-[5 pts.]- Sketch x(t), y(t) and [x(t), y(t)].
CONTINUED ON NEXT PAGE
C. O. Bloom
306T&Y
TEST #2 - Study Guide
Problem # 4:
Euler's Method
See section # 2.4 of Text #1
(a)-[15 pts.]- Use Euler's method with step size ∆t = # to approximate the solution
of the following initial value problem: x' = f(x, y) , y' = g(x, y), x(0) = x0, y(0) = y0
over the interval [a, b].
(b)-[5 pts.]- Plot the points (xi, yi ) found by Euler's method.
Draw a polygonal arc through these points.
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Problem #5:
Solutions of Linear Systems
See problems 26-29 in section 3.1 of Text #1.
v
v
(a)-[5 pts.] - Verify that Y (t)  e 1t  11 and Y (t)  e  2 t  12  are solutions of Y '  AY .
1
2


v21 

v22 

(b)-[5 pts.]- Verify that Y1(0) and Y2 (0) are linearly independent, i.e. set k1Y1 (0)  k2Y2 (0)  0
and show that k1  0 and k 2  0 .
(c)-[5 pts.]- Write down the general solution of Y '  AY .
x 
(d)-[5 pts.]- Find k1 and k2 such that Y(t)  k1Y1(t) + k2Y2(t) satisfies the initial condition Y (0)   0  .
 y0 
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Problem #6:
An initial value problem for a first order linear system of o.d.e's
See Section 3.2, pp. 259-261, and Section 3.2. Do problems #1, 3, 5, 11, 13 on pg. 264.
Problem on actual test will be similar to the one here described but with different numerical coefficients.
(a)-[4 pts.]- Rewrite the following system in matrix-vector form: x'  x  y, y'  3x  y .
(b)-[5 pts.]-Find a linearly independent pair of vector solutions Y1(t), Y2 (t) of the given system that
v 
have the form Y (t )  e tV , V =  1 
v 2 
(c)-[5 pts.]-Use the results of (b) to construct the general solution of the given system.
 2
(d)-[3 pts.]-Find the values of the arbitrary constants in the general solution so that Y (0)    .
1 
(e)-[3 pts.]-Use information from part (b) to classify the equilibrium point (0, 0) of the given system.
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Problem # 7:
An Overdamped Harmonic Oscillator Problem
See section 3.1, problem 19. See also “A Harmonic Oscillator” on page 261 and “An Overdamped Harmonic
Oscillator” on page 325. See section 3.6 pp. 316-321. Do problems (3.6) #1, 5, 7, 9, 11.
(a)-(i)-[5 pts.]- Re-write the differential equation ay ' 'by ' cy  0 as an equivalent 1st order system of two linear
o.d.e’s in two unknowns. Here a, b and c are positive constants.
(a)-(ii)-[5 pts.]- Write the system of part (a) in matrix-vector notation, i.e. in the form Y '  AY where A is a 2x2
matrix and Y is a 2x1 vector.
(b)-(i)-[5 pts.]- Find the general solution of the system found in part (a)-(ii).
(c)-[5 pts.]- Use the result of (b)-(i) to find the general solution of ay ' 'by ' cy  0 .
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C. O. Bloom