calculus homework

MAT 17 C — Luis Rademacher Midterm 2 — May 22nd — 50 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

dx1 dt

= 3x1 � 2x2 dx2 dt

= �1x1 + 4x2

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 � 0.5x2.

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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.)

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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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