Final Exam

Question 1

Ordinary differential equation (ODE)

Answer 1

unknown function x has one independent variable t. Equation relates x and its derivatives to values of t.

Question 2

First order linear ODE

Answer 2

Question 3

How to solve?

Answer 3

integrate both sides n-times, will have n arbitrary constants

Question 4

How to solve?

Answer 4

separation of variables

Question 5

How to solve?

Answer 5

first-order linear ODE (not separable)

Question 6

Example: Assume that a drug is absorbed by the body at a rate proportional to the amount of the drug present in the bloodstream at time 𝑡. Let’s write 𝑥(𝑡) for the mg of drug present in the bloodstream at time 𝑡.

Find an ODE for which 𝑥(𝑡) is a solution, assuming that the drug is absorbed at a rate of 0.4𝑥(𝑡) per hour and that the patient is simultaneously receiving the drug intravenously at the constant rate of 10 mg/hour:

Answer 6

Question 7

N-th order ODE is linear

Answer 7

functions in front of the derivatives only depend on t

Question 8

N-th order ODE is HOMOGENOUS

Answer 8

E(t) = 0

Question 9

N-th order ODE is NORMAL

Answer 9

leading coefficient not equal 0

continuous coefficients

Question 10

General solution to a linear ODE

Answer 10

x = p(t) + H(t)

Question 11

What is the Existence & uniqueness theorem for ODEs, applies to initial value problems

Answer 11

Question 12

Operator

Answer 12

Question 13

What does it mean that linear differential operators are linear?

Answer 13

1. L[x1 + x2] = Lx1 + Lx2
2. L[kx] = k Lx

Question 14

Cramer’s rule to solve Bx = A

Answer 14

Replace the ith column of B with the vector A, take determinant.

Question 15

Linear combination of homogenous solution L[x] = 0

Answer 15

Also a homogenous solution: L[c1h1(t) + c2h2(t)] = 0

If you found n linearly independent solutions to a n’th order ode, then any other solution can be written as a linear combination of those

Question 16

Existence & uniqueness theorem for nth order linear ODEs

Answer 16

If you have n initial conditions, then x = x(t) has a UNIQUE solution for a fixed value t0.

Question 17

Linearly dependent vs independent

Answer 17

Question 18

Wornskian test for linear independence:

Answer 18

Question 19

Theorem: Wronskian for solutions, non-vanishing never vanishing:

Answer 19

Question 20

Exponential shift formula

Answer 20

Question 21

Euler’s formula

Answer 21

Question 22

How to solve a homogeneous linear equation with constant coefficients:

Answer 22

Question 23

Find a particular solution p(t) to linear ODE, P(D)x = E(t)

Answer 23

Question 24

Find equivalent system of ODEs in matrix form: Dx = Ax + E

Answer 24

Question 25

Answer 25

Question 26

Eigenvector

Answer 26

(A – λI)x = 0

Av = λv

Question 27

Eigenvalue

Answer 27

det(A – λI) ≠ 0

Question 28

What is the solution to Dx = Ax if λ is an eigenvalue of A with corresponding eigenvector v.

Answer 28

Question 29

If A is an (n x n) matrix with n distinct eigenvalues

Answer 29

The eigenvectors must be linearly INDEPENDENT

Question 30

How to easily check if vectors are linearly independent?

Answer 30

Make a matrix constructed of column vectors. The vectors are linearly independent if

det ≠ 0

Question 31

When solving non-homogenous matrix equations via row operations & row reduction

Answer 31

variables – # equation = # free variables

Question 32

Homogeneous linear system

Answer 32

Ax = 0

Question 33

Homogeneous linear system (Dx = Ax) constant coeff, COMPLEX ROOTS. How to solve?

Answer 33

Question 34

If square matrix A has eigenvalue with multiplicity d>1, d is the max # linearly independent-eigenvectors

Answer 34

Less than d eigenvectors cannot describe general solution

Question 35

Generalised eigenvector

Answer 35

Question 36

How to find the solution to Dx = Ax associated with a generalized eigenvector

Answer 36

Question 37

Answer 37

Question 38

Find general solution to Dx = Ax + E, VARIATION OF PARAMETERS

Answer 38

Question 39

How to draw phase diagrams?

Answer 39

1. Plot eigenvectors (lines)
2. Add directions (λ>0 arrow away from origin)

Question 40

Partial fraction decomposition

Answer 40

Question 41

Rewrite the function 𝑓(𝑡) using the unit step functions

Answer 41

Question 42

Integral curve vs. phase portraits:

Answer 42

Integral curves: solutions to linear system Dx=Ax

Phase portraits: all integral curves together with arrows for direction

Question 43

Characterizing phase portraits using eigenvalues:

2 real eigenvalues (OPPOSITE SIGN) with corresponding eigenvectors v, w

Answer 43

two lines through the origin determined by v,w.

Every other curve asymptotic to line corresponding to the smaller λ as t → –∞, and to line corresponding to the larger to the larger λ as t → ∞.

Question 44

Characterizing phase portraits using eigenvalues:

2 real eigenvalues (SAME SIGN) with corresponding eigenvectors v, w

Answer 44

curves have slopes approaching the slopes of these lines

Question 45

Characterizing phase portraits using eigenvalues:

1 real eigenvalue with corresponding eigenvector v

Answer 45

(multiplicity 2) 2 linearly independent eigenvectors: curves are radial lines through the origin

1 eigenvector: slopes approaching the slope of this line as t→ + or –∞

Question 46

Characterizing phase portraits using eigenvalues:

complex eigenvalues

Answer 46

Question 47

Autonomous system

Answer 47

Question 48

Equilibrium

Answer 48

constant solution of the form x(t) = c

Equilibria occur at points c where the derivatives of the coordinates vanish: dx/dt(a,b) = dy/dt(a,b) = 0

Question 49

Properties of equilibrium:

Eigenvalues of A have NEGATIVE real part

Answer 49

c is an attractor (stable)

Question 50

Properties of equilibrium:

Eigenvalues of A have POSITIVE real part

Answer 50

c is a repeller (unstable)

Question 51

Properties of equilibrium:

Eigenvalues with real parts of opposite sign, zero, or purely imaginary

Answer 51

c is neither attractor or repeller (unstable)

Question 52

Linearisation matrix

Answer 52