306D
TEST #2 - Study Guide
Do any three problems. Any of the remaining 3 problems may be done for extra credit.
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Problem # 1:
Euler's Method
(a)-[15 pts.]- Use Euler's method with step size ∆t = # to approximate the solution
of the following initial value problem: x' = f(t, x), x(0) = x0 for a  t  b .
(b)-[10 pts.]- Plot the points (xi, yi ) found by Euler's method and draw a polygonal arc
connecting these points.
(See section # 6.1, pp. 301-304. Do probs. #1, 3, 7, 9, 19 pp. 311-312. )
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Problem 2:
(a)-[5 pts.] Write down the augmented matrix of a given system of three linear, algebraic equations in
in three unknowns.
(b)-[10 pts.] Find the reduced row echelon form of the augmented matrix found in part (a).
(c)-[5 pts.] Use the result of part (b) to find the solution set for the given algebraic system.
(See section 7.3, especially “Reduced Row Echelon Form” on p. 360. Do probs. 19, 23 on pg. 361-2.)
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Problem 3:
Consider the linear, algebraic system Ax=b, where
 a11 a12 a13 
 x


 
A   a21 a22 a23  and
x =  y.
a

z
 31 a32 a33 
 
(a)-[15 pts. ] Find A-1 .
(b)-[10 pts.] Calculate A -1b . This is the vector solution of Ax=b.
(See “Invertible Matrices”, pp. 368. Do probs. 11, 20, 21 on pp. 369-370. See “Definition of a Determinant on
pp. 386-390, especially propositions 6.11, 6.12, 6.13, 6.14 on p. 389. See Ex. 6.15, 6.16 on p. 390. Do probs. 7,
9, 11 on p. 392.)
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306D
TEST #2 - Study Guide
Problem# 4
First Order Systems of Differential Equations
(a)-[8 pts.] Show that given functions x(t) and y(t) satisfy a given system of first order differential equations,
of the form x '  f ( x, y ), y '  g ( x, y ) with given initial conditions x(0)  x0 , y(0)  y0 .
(b)-[8 pts.]-Write a given initial value initial value problem of the form
x '' ax ' bx  f (t ), x(0)  x0 , x '(0)  v0 as an initial value problem for a first order system
of differential equations in vector notation.
(c)-[9 pts.]- Write a given initial value initial value problem of the form
y '''  f ( y, y ', y ''), y (0)   , y '(0)   , y ''(0)   as an initial value problem for a first
order system of differential equations in vector notation.
(See section 8.3. Do probs. 7-9, 11, 13, 15, 16 on p. 402.)
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C. O. Bloom
Problem #5:
A Predator Prey Model
(a)-[15 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)-[5 pts.]- Sketch a typical solution of the predator-prey system you obtained in part (a) on
Figure 1 below, where W(t) stands for the Wolf population at time t  0, and M(t) stands for
the mouse population at time t  0.
(c)-[5 pts.]- Sketch the graphs of W(t) and M(t) on Figure 2 below. Briefly describe the
behavior of W(t) and M(t) as functions of t.
See section 8.2, pp. 405-410. Do prob. # 26 on p. 416 in Text #1
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Problem #6:
Qualitative Behavior of Solutions of autonomous Systems
(a)-[5 pts.]- Locate and plot the equilibrium points of a given non-linear system of the form
x'  f (x, y), y '  g(x, y) .You may assume that f (x, y) and g(x, y) are polynomials in x and y .
(b)-[10 pts.]- Find the x and y nullclines for the given system, and sketch them.
(c)-[5 pts.]-Sketch the direction field of the given system along each nullcline.
(f)-[5 pts.]-Sketch the direction field of the given system along the phase plot of a given solution. system.
(See section 8.2, pp. 405-410 & 8.3, pp. 420-422. Do probs. 1, 2, 3, 12, pp. 423-424.)
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