I need help with calculus homework

MAT17 C — Luis Rademacher

Midterm

2

— May 22nd —

5

0 minutes

Open book and open notes but every student must write his or her own solution. Justify

or prove all your answers.

1. Solve the system of ODEs

dx

1

dt

=

3

x1 � 2×2

dx2

dt

= �1×1 +

4

x2

with initial condition x1(0) = 1, x2(0) = 1.

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2. Find the corresponding compartment diagram (determine a, b, c, d) for the following

system of di↵erential equations. Then analyze the stability of equilibrium (0, 0).

dx1

dt

= �x1

dx2

dt

= x1 � 0.5×2.

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3. Find all equilibria of the following system of di↵erential equations and use the analytical

approach to determine the stability of each equilibrium:

dx1

dt

= x1x2 � x2

dx2

dt

= x1 + x2.

(Analytical approach = solve system of non-linear equations to find equilibria if needed,

compute Jacobi matrix for linearization and study eigenvalues.)

3

4. Assume that

dx1

dt

= x1(2 � x1) � x1x2

dx2

dt

= x1x2 � x2.

(a) Graph the zero isoclines.

(b) Show that (1, 1) is an equilibrium and use the analytical approach to determine

its stability.

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5. (a) Forty patients wish to enroll in a small study in which patients are divided into

four groups of 10 patients each. In how many ways can this be done if no patient

is to be assigned to more than one group?

(b) Toss five fair coins and find the probability of three or more heads.

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