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Maths > Dr Musa > Flashcards

Dr Musa Flashcards

1
Q

What are differential equations?

A

A differential equation is an algerbraic or transcendental equation which involves derivatives

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2
Q

What are transcendental equations?

Give some examples

A

These are equations containing transcendental functions i.e. they transcend algebra. They cannot be expressed in terms of finite sequence of the alegebraic operations of addition, multiplication and root extraction.

Expontential functions, (ex, xπ) logarithmic (log (x)) and trionometric functions (sin(x), cos(x)) are example of transcendental functions

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3
Q

What are difference equations

A

A difference equation is an algebraic or transcendental equation which involves a dependent variable, say y(k), defined on a discrete set of the independent variables, say x(k), where k is the integer-valued discrete time variable

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4
Q

Differential equations relate to ____-____ ____ in the same sense as difference equations relate to ____-____ ____

A

Differential equations relate to continuous-time systems in the same sense as difference equations relate to discrete-time systems

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5
Q

What is an ordinary differential equation?

A

An equation involving;

  1. One independent variable
  2. One or more dependant variable, and
  3. One or more derivatives of the dependent varable with respect to the indedpendent variable
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6
Q

What is a partial differential equation?

A

An equation involving;

  1. Two or more independant variables
  2. One or more dependent varibles, and
  3. One or more partial derivatives of the dependent variable (or variables) with respect to the independent variable
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7
Q

What is a term of a differential equation?

A

An explicit function of the independent variable, the dependent variable and the derivatives of the dependent variable

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8
Q

What is a term of a difference equation?

A

A product and/or quotient of an explicit function of the independent variable and the dependent variable

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9
Q

What is a linear term in a differential equation?

A

A term whose degree in the dependent variable or their derivatives is 1 (i.e. raised to the power - or degree - or 1)

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10
Q

What is a linear differential equation?

A

A differential equation consisting of only linear terms

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11
Q

The class of linear differential equations where ai and bi are constants is called…

and these equations represent…?

A

…Linear differential equations with constant co-efficients and these equations represent linear time-invarient systems

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12
Q

y=y(t) is often called the ____ or the ____ of the system, and is the ____ solutions fo the differential equation to be determined.

Whereas u=u(t) is often called the ____, and is a ____ (or ____) function

A

y=y(t) is often called the output or the response of the system, and is the unknown solution fo the differential equation to be determined.

Whereas u=u(t) is often called the input, and is a known (or given) function

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13
Q

For physical systems, m ? n and n is the ____ of the diff eqn

A

m ≤ n and n is the order of the diff eqn

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14
Q

The class of linear difference equation where ai and bi are constants is called…?

A

…linear difference equations with constant coefficients

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15
Q

Diff/diffence equations that are not linear are called…?

A

…non linear differential/difference eqns

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16
Q

What is the order of a diff eqn?

A

The order of a diff equation is the order of the highest derivative appearing in it

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17
Q

What is the order of a difference eqn?

A

The difference between the largest and the smallest arguements of the interger-valued discrete variable appearing in it

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18
Q

What is the degree of a diff eqn?

A

The degree of a diff eqn that can be written as a polynomial, in the derivatives is the degree of the highest ordered derivatives that appear in it

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19
Q

What is a differential operator?

A

When you replace d/dt with a symbol or a letter, say D

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20
Q

An equation or polynomial containing a differential operator is called…?

A

…the auxiliary or characteristic eqn

D3+D2+2=0

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21
Q

Linear differential equation of the form:

is called a…?

A

homogeneous nth-order linear differential equation if f(t) = 0; otherwise it is non homogeneous

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23
Q

dy/dt + ry = ? in terms of D

and so D = ?

And then this solution can be wirtten as;

y = ?

A

(D+r)y = 0

D = -r

y = CeDt

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24
Q

If D

Therefore, the system is ____

A

If D decays as t…tends to infinity.

Therefore, the system is stable

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25
Q

If D

Therefore, the system is ____

A

If Dincreases unboundelly as t tends to infinity.

Therefore, the system is unstable

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26
Q

If the (b2-4ac) > 0, the roots of the characteristic equation D1 and D2 are ____ and ____ (i.e. ____)

With the general solution being;

y = ? + ?

A

If the (b2-4ac) > 0, the roots of the characteristic equation D1 and D2 are real and distinct (i.e. unequal)

With the general solution being;

y = c1eD1t + c2eD2t

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27
Q

If (b2-4ac) = 0 then the roots of the characteristic are ____ and ____

The general solution is

y = ? + ?

A

If (b2-4ac) = 0 then the roots of the characteristic are real and equal

The general solution is

y = c1eD1t + c2teD2t

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28
Q

If (b2-4ac)

The general solution is

y = ? + ?

A

If (b2-4ac) complex numbers

The general solution is

y = K1eD1t + K2e<span>D2t</span>

Using Eulier’s identity the last expression can be simplified to

y = eøt(c1 cosßt + c2 sinßt)

Where c1 = K1 + K2

c2 = j(K1 - K2)

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29
Q

2nd-order systems have _ inital conditions, an nth-order system will have _ inital conditions

A

2nd-order systems have 2 inital conditions, an nth-order system will have n inital conditions

30
Q

y’’’ - y’ = 0

It’s characteristic equation is…?

A

D3 - D = 0

31
Q

Define steady state response

A

The point of the total response that does not approach zero as time approaches infinity

32
Q

Define transient response

A

The point of the total response that approaches zero as time approaches infinity

33
Q

y(t) = ya(t) + yb(t)

What is the forced response and what is the free response?

A

y(t) = ya(t) + yb(t)

ya is the free response

yb is the forced response

34
Q

yb(t) = ∫ w(t-J)

What is w(t-J) called?

A

The weighting function or the kernal of the differential equation

35
Q

This is provided in the data booklet, but what would the Laplace transform of a single-sided or unilateral look like?

Where would it be useful?

A

The change in ∞ to 0t

This would be particularly useful for finding the Laplace transform of functions that are discontinuous at t=0

37
Q

In Laplace transform, L is called the…?

A

Laplace transform operator

38
Q

What is Euler’s Identity?

A

eajt = cos(at) + j sin(at)

39
Q

Time scaling: If the Laplace transform of a function x(t) is x(s), then the Laplace transform of the function x(at) is given by…

A

L{x(at)] = 1/a . X(s/a)

40
Q

If you have an equation like this, what should you do?

A
  1. Factorise
  2. Partial fraction
41
Q

Division by t: If the Laplace transform of a function x(t) is X(s), then the Laplace transform of x(t)/t is given by…

A

L{x(t)/t}= ∞∫s X(u)du

42
Q

Frequency scaling property: If x(s) is the inverse Laplace transform of x(t), then the inverse Laplace transform of x(ks) is…?

A

1/k.x(t/k)

43
Q

What is the complex convolution intergral of x1(t) and x2(t) ?

A
44
Q

The first term of on the right of the equation is the ____ ____ and the second term is the ____ ____ of the system

A

The first term of on the right of the equation is the forced response and the second term is the free response of the system

45
Q

The transfer function of a LTI system is the point of the first term in the right side of the equation multiplying U(s)

A
46
Q

Define the transfer function of a LTI system

A

The ratio of the Laplace transform of the output variable Y(s) to the Laplace transform of the input variable U(s), with all initial conditions assumed to be zero

47
Q

The output of any continuous-time LTI system is the…?

A

…convolution of the input with the impulse response of the system

48
Q

What are the roots of the characteristic equation called?

A

Poles

49
Q

What are the roots of numerator polynomial of the transfer function called?

A

Zeros

50
Q

When is the system stable?

A

If all of the roots of the characteristic equation (that is the system poles) have negative real point/parts

51
Q

The transfer function, G(s) = ?

A

G(s) = Y(s) / U(s)

52
Q

What is the characteristic equation in this equation?

A

Characteristic equation: s2+4s+3

54
Q

The characteristic equation is the…?

A

…denominator

55
Q

If the input is a unit step function, u(t) = 1, so

u(s)?

A

u(s) = L{u(t)} = 1/s

56
Q

If the input is a unit impluse function, ∂(t) = u(t) = ?

A

∂(t) = u(t) = 1

57
Q

What are Fourier seires?

A

Fourier series are infinite series designed to represent general periodic functions in terms of cosines and sines

58
Q

In the Fourier series, wo = ?

And To = ?

A

wo = 2π/To

To = 2L

59
Q

In Fourier series, f(t) = ?

A
60
Q

The Fourier coefficients given by the so-called Euler formulas

an = ?

bn = ?

A
61
Q

In the Fourier series,

a0 = ?

A
62
Q

The graph of an even function is symmetric with respect to the…?

A

…y-axis

63
Q

The graph of an odd function is symmetric about the…?

A

…origin

64
Q
A
65
Q

The intergral of an even function over a symmetric interval (-a,a) is ____ the intergral of that function over (0,a), that is, if f(t) is an ____ function then…

A

The intergration of an even function over a symmetric interval (-a,a) is twice the intergral of that function over (0,a), that is, if f(t) is an even function then…

66
Q

The intergral of an ____ function over a symmetric interval (-a,a) is zero. If g(t) is an ____ function then…

A

The intergral of an odd function over a symmetric interval (-a,a) is zero. If g(t) is an odd function then…

67
Q

The Fourier series of an even function is a “Fourier ____ series” because the Gourier coefficients bn are all identically equal to ____

A

The Fourier series of an even function is a “Fourier cosine series” because the Gourier coefficients bn​ are all identically equal to zero

68
Q

The Fourier series of an odd function is a “Fourier ____ series” because the Fourier coefficients an, including a0, are all identically equal to ____

A

The Fourier series of an odd function is a “Fourier sine series” because the Fourier coefficients an, including a0, are all identically equal to zero

69
Q

What is the equation for the avergae power P of a periodic signal x(t) over any period?

A

where denotes the fundamental period of x(t)

70
Q

What is Parseval’s Identity for Fourier series?

A
71
Q

State Parselva’s Identity with proof

A

If an and bn are the Fourier coefficient components dying down to f(x) and if f(x) satifies the Dirichelet conditions

Maths (4 decks)

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