I need help with calculus homework
MAT17 C — Luis Rademacher
— May 22nd —
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
x1 � 2×2
= �1×1 +
with initial condition x1(0) = 1, x2(0) = 1.
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).
= x1 � 0.5×2.
3. Find all equilibria of the following system of di↵erential equations and use the analytical
approach to determine the stability of each equilibrium:
= x1x2 � x2
= x1 + x2.
(Analytical approach = solve system of non-linear equations to find equilibria if needed,
compute Jacobi matrix for linearization and study eigenvalues.)
4. Assume that
= x1(2 � x1) � x1x2
= x1x2 � x2.
(a) Graph the zero isoclines.
(b) Show that (1, 1) is an equilibrium and use the analytical approach to determine
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.