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MATH 2351
Ordinary Differential Equations
Spring 2013
Quiz 2
1. 8 points Consider the following predator-prey system:
dx
xy
= −x +
,
dt
2
dy
= 2y(7 − y) − 5xy.
dt
(a) Identify which variable represents the predator population, and which one represents the
prey population. Which ones are larger, the predators or the preys?
(b) Find all the equlibrium solutions of this system.
2. 8 points Conside the following (partially decoupled) system:
(a) Find the general solution of this system.
(b) Find the solution satisfying x(0) = 1, y(0) = 0.
dx
= −4x,
dt
dy
= 2x + 3y.
dt
MATH 2351
Quiz 2
Spring 2013
3. 4 points
(a) Convert the differential equation y 00 (t) = 3y 0 (t) + 8y(t) − 5y 2 (t) to a system of first order
differential equations. You DO NOT have to solve the system.
(b) Write the system
dx1
dx2
= 2x1 − 7x2 ,
= 9x1 + x2 in matrix form.
dt
dt
3 2
.
6 −1
4. 10 points Consider the linear system dX/dt = AX, where A =
−1
1
−3t
5t
(a) Verify that X1 (t) = e
and X2 (t) = e
are solutions to this system.
3
1
(b) Verify that X1 (0) and X2 (0) are linearly independent vectors in the plane.
0
(c) Find the solution X(t) with the initial value X(0) =
.
4