Chapter 4 (c)

Question 1

Second Order ODE
General Form (independent variable x, dependent variable, t)

Answer 1

d²x/dt² + a dx/dt + bx = h(t)

Question 2

Second Order ODE Written in Terms of First Order Differentials

Answer 2

x’’ +ax’ + bx = h(t)

y = x’ , so y’ = x’’
this gives
y’ = h(t) - ay - bx

Question 3

Two Dimensional First Order ODE - General

Answer 3

(x') = (0   1) (x) + ( 0  )
(y') = (-b -a) (y) + (h(t))

Question 4

Two Dimensional First Order ODE - Autonomous Inhomogeneous Constant Coefficient ODE

Answer 4

(x’ y’) = A (x y) + (H1 H2)

where H1 and H2 are constants

Question 5

Two Dimensional First Order ODE - Autonomous Homogeneous Constant Coefficient ODE

Answer 5

(x’ y’) = A(x y)

Question 6

The Phase Plane - General Approach

Answer 6

-the general approach to two-dimensional autonomous ODEs is to interpret the independent variable, t, as time and consider the time-dependent point (x(t) , y(t)) in the x-y plane

Question 7

The Phase Plane - Trajectory

Answer 7

-the point (x(t) , y(t)) traces a curve in the phase plane called the trajectory of the initial condition (x0 , y0)

Question 8

The Phase Plane - Phase Portrait

Answer 8

-a diagram showing one or more trajectories

Question 9

The Phase Plane - Equilibrium Points

Answer 9

-a point with (x’,y’) = (0,0) is called an equilibrium point

Question 10

The Phase Plane - Ordinary Points

Answer 10

-a point that is not an equilibrium point (i.e. most points on the phase plane)

Question 11

The Phase Plane - Theorems for Autonomous ODEs

Answer 11

• there is only one trajectory through each ordinary point
• any trajectory starting at a ordinary point cannot reach an equilibrium point in a finite amount of time
• a trajectory through an ordinary point cannot cross itself unless it is a closed curve, in which case the trajectory is periodic

Question 12

The Phase Plane - Eigenvalues

Answer 12

• find the eigenvalues of matrix A
• the nature of the solution to the equation depends entirely on the roots of the characteristic equation, the eigenvalues of A

Question 13

Classifying Equilibrium Points

Stable Node

Answer 13

real eigen values that are unequal and negative

Question 14

Classifying Equilibrium Points

Unstable Node

Answer 14

-real eigenvalues that are unequal and positive

Question 15

Classifying Equilibrium Points

Saddle

Answer 15

real eigenvalues that have opposite signs

Question 16

Classifying Equilibrium Points

Centre

Answer 16

complex eigenvalues with zero real part

Question 17

Classifying Equilibrium Points

Stable Focus

Answer 17

-complex eigenvalues with negative real part

Question 18

Classifying Equilibrium Points

Unstable Focus

Answer 18

-complex eigenvalues with positive real part

Question 19

Classifying Equilibrium Points

Special Cases

Answer 19

• eigenvalues are equal (there is only one eigenvector) this case is called an improper node
• one or both eigenvalues are zero

Question 20

Writing the Solution of the ODE in the Case of Real Unequal Eigenvectors

Answer 20

Av1 = λ1v1 and Av2 = λ2v2

(x y) = C1e^(λ1t)v1 + C2e^(λ2t)v2

Question 21

Drawing a Phase DIagram

Answer 21

• draw the lines represented by the eigenvectors
• if the eigenvalue corresponding to the eigenvector is positive then the line points away from the equilibrium point, if it is negative then the line points towards the equilibrium point