##### See questions in the file, please answer them using the book-(Answered)

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See questions in the file, please answer them using the book "elementary differential equations"

M344 Final

Siddharth Bhaskar

1. Let C1 be the circle of radius 1 centered around (2, 0) and let C2 be the

circle of radius 1 centered around (?2, 0). Let C3 be the circle of radius

4 centered around the origin. Notice that C1 and C2 lie in the interior

of C3 . Let R be the region of the plane inside C3 but outside C1 and

C2 . For each of the following possibilities, either draw the trajectories of

the smooth vector eld indicated, or argue that no such eld exists. (A

picture will be drawn during the exam.)

(a) A vector eld with no critical points in R but with a periodic solution

lying on each of the circles. (5 points)

(b) A vector eld with exactly one critical point in R and with a periodic

solution lying on each of the circles. (5 points)

2. Suppose R is a region of the plane that captures some trajectory of a

smooth vector eld F. Suppose all the critical points of F inside R are

unstable. Using the Poincare-Bendixson theorem, argue that R contains

a periodic solution of F. (10 points)

3. The functions cosh and sinh are dened by

cosh t =

et ? e?t

et + e?t

, sinh t =

.

2

2

(a) Show that if {(et , et ), (e?t , ?e?t )} is a fundamental set of solutions,

then so is {(cosh t, sinh t), (sinh t, cosh t)}. (5 points)

(b) Find a matrix A such that {(cosh t, sinh t), (sinh t, cosh t)} is a fundamental set of solutions to x0 = Ax. (5 points)

4. Let x be a function of t and f and g be smooth functions of x. Suppose

x satises the second-order nonlinear dierential equation

d2 x

dx

+ f (x)

+ g(x) = 0.

2

dt

dt

Answer the following questions. Your answers will be in terms of f and g .

(a) Rewrite this equation as a rst-order nonlinear system of dierential

equations. In other words, let y = dx

dt , and nd a vector eld F such

dx dy

that ( dt , dt ) = F(x, y). (3 points)

1

(b) Compute the Jacobian of F. (3 points)

(c) Suppose (0, 0) is a critical point of F. Evaluate the Jacobian at (0, 0)

and compute the trace and determinant. Your answers should be in

terms of f (0) and g(0). (4 points)

5. Consider the following family of nonlinear systems parameterized by ? :

x0 = ? ? y

y 0 = ?4x + y + x2 .

(a) Give conditions on ? which cause this system to have zero, one, or

two critical points. (5 points)

(b) In the case that this system has two critical points, show that one of

them must be a saddle point. (5 points)

6. Let A be a 2x2 matrix. Suppose that A has trace 10 and determinant 9.

(a) Calculate the eigenvalues of A. (5 points)

(b) What type of critical point is the origin in the equation x0 = Ax?

Justify your answer. (5 points)

7. The equations x(t) = (a cos t, b sin t) parameterize an ellipse. Find a matrix A such that x(t) is a solution to x0 = Ax. (10 points)

8. Consider the competing species equation

dx

= x(1 ? x ? y)

dt

dy

= y(? ? 2y ? x)

dt

parameterized by ?. Notice that (1, 0) is a critical point.

(a) Graph the nullclines assuming 1 < ? < 2 and, in this case, determine

the type of critical point of (1, 0). (5 points)

(b) Do the same assuming ? < 1. (5 points)

9. Find a fundamental set of solutions to the rst-order system

?3

0

1

x =

?2

(10 points)

2

0 2

?1 0 x

?1 0

10. Consider the system of dierential equations in polar coordinates

dr

= r(r ? 1)(r ? 2)

dt

d?

= 1.

dt

Describe all periodic solutions of this system along with their stability

characteristics, in other words, whether they're stable, asymptotically stable, or unstable, on the inside and outside. (10 points)

11. Find a 2x2 real matrix A such that (e?4t , 0) and (e?4t , te?4t ) form a

fundamental set of solutions to x0 = Ax. (10 points)

12. Consider the predator-prey equation

y

dx

= x(1 ? )

dt

2

dy

x

= y(?1 + )

dt

3

(a) Find the unique critical point not at the origin, and evaluate the

Jacobian at this critical point. (5 points)

(b) Find a fundamental set of solutions to x0 = Jx, where J is the matrix

from part (a). (5 points)

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