Dr Musa

Question 1

What are differential equations?

Answer 1

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

Question 2

What are transcendental equations?

Give some examples

Answer 2

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, (e^x, x^π) logarithmic (log (x)) and trionometric functions (sin(x), cos(x)) are example of transcendental functions

Question 3

What are difference equations

Answer 3

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

Question 4

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

Answer 4

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

Question 5

What is an ordinary differential equation?

Answer 5

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

Question 6

What is a partial differential equation?

Answer 6

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

Question 7

What is a term of a differential equation?

Answer 7

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

Question 8

What is a term of a difference equation?

Answer 8

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

Question 9

What is a linear term in a differential equation?

Answer 9

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

Question 10

What is a linear differential equation?

Answer 10

A differential equation consisting of only linear terms

Question 11

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

and these equations represent…?

Answer 11

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

Question 12

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

Answer 12

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

Question 13

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

Answer 13

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

Question 14

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

Answer 14

…linear difference equations with constant coefficients

Question 15

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

Answer 15

…non linear differential/difference eqns

Question 16

What is the order of a diff eqn?

Answer 16

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

Question 17

What is the order of a difference eqn?

Answer 17

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

Question 18

What is the degree of a diff eqn?

Answer 18

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

Question 19

What is a differential operator?

Answer 19

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

Question 20

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

Answer 20

…the auxiliary or characteristic eqn

D³+D²+2=0

Question 21

Linear differential equation of the form:

is called a…?

Answer 21

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

Question 23

dy/dt + ry = ? in terms of D

and so D = ?

And then this solution can be wirtten as;

y = ?

Answer 23

(D+r)y = 0

D = -r

y = Ce^Dt

Question 24

If D

Therefore, the system is ____

Answer 24

If D decays as t…tends to infinity.

Therefore, the system is stable

Question 25

If D

Therefore, the system is ____

Answer 25

If Dincreases unboundelly as t tends to infinity.

Therefore, the system is unstable

Question 26

If the (b²-4ac) \> 0, the roots of the characteristic equation D₁ and D₂ are ____ and ____ (i.e. \_\_\_\_) With the general solution being; y = ? + ?

Answer 26

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 = **c₁e^D₁t + c₂e^D₂t**

Question 27

If (b²-4ac) = 0 then the roots of the characteristic are ____ and \_\_\_\_ The general solution is y = ? + ?

Answer 27

If (b²-4ac) = 0 then the roots of the characteristic are **_real_** and **_equal_** The general solution is y = **c₁e^D₁t + c₂**_t_**e^D₂t**

Question 28

If (b²-4ac) The general solution is y = ? + ?

Answer 28

If (b²-4ac) _complex numbers_ The general solution is y = K₁e^D₁t + K₂e^D₂t Using **Eulier's identity** the last expression can be simplified to y = e^øt(c₁ cosßt + c₂ sinßt) Where c₁ = K₁ + K₂ c₂ = j(K₁ - K₂₎

Question 29

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

Answer 29

2nd-order systems have **_2_** inital conditions, an nth-order system will have **_n_** inital conditions

Question 30

y''' - y' = 0 It's characteristic equation is...?

Answer 30

D³ - D = 0

Question 31

Define **steady state response**

Answer 31

The point of the total response that **_does not_** approach **zero** as time approaches **infinity**

Question 32

Define **transient response**

Answer 32

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

Question 33

y(t) = y_a(t) + y_b(t) What is the **forced response** and what is the **free** **response**?

Answer 33

y(t) = y_a(t) + y_b(t) y_a is the **free response** y_b is the **forced response**

Question 34

y_b(t) = ∫ w(t-J) What is w(t-J) called?

Answer 34

The **weighting function** or the **kernal** of the differential equation

Question 35

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?

Answer 35

The change in ∞ to 0t This would be particularly useful for finding the Laplace transform of functions that are **discontinuous** at t=0

Question 37

In Laplace transform, L is called the...?

Answer 37

**Laplace transform operator**

Question 38

What is Euler's Identity?

Answer 38

e^ajt = cos(at) + j sin(at)

Question 39

**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...

Answer 39

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

Question 40

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

Answer 40

1. Factorise 2. Partial fraction

Question 41

**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...

Answer 41

L{x(t)/t}= ∞∫s X(u)du

Question 42

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

Answer 42

1/k.x(t/k)

Question 43

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

Answer 43

Question 44

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

Answer 44

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

Question 45

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

Question 46

Define the **transfer function** of a LTI system

Answer 46

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**

Question 47

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

Answer 47

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

Question 48

What are the **roots** of the characteristic equation called?

Answer 48

**Poles**

Question 49

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

Answer 49

**Zeros**

Question 50

When is the system stable?

Answer 50

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

Question 51

The **transfer function**, G(s) = ?

Answer 51

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

Question 52

What is the **characteristic equation** in this equation?

Answer 52

**Characteristic equation**: s²+4s+3

Question 54

The characteristic equation is the...?

Answer 54

...denominator

Question 55

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

Answer 55

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

Question 56

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

Answer 56

∂(t) = u(t) = 1

Question 57

What are **Fourier seires?**

Answer 57

Fourier series are **infinite** **series** designed to represent **general** **periodic** fu**n**ctions in terms of **cosines** and **sines**

Question 58

In the Fourier series, w_o = ? And T_o = ?

Answer 58

w_o = 2π/T_o T_o = 2L

Question 59

In Fourier series, f(t) = ?

Answer 59

Question 60

The Fourier coefficients given by the so-called Euler formulas a_n = ? b_n = ?

Answer 60

Question 61

In the Fourier series, a₀ = ?

Answer 61

Question 62

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

Answer 62

...y-axis

Question 63

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

Answer 63

...origin

Question 64

Answer 64

Question 65

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...

Answer 65

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...

Question 66

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

Answer 66

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

Question 67

The Fourier series of an even function is a "Fourier ____ series" because the Gourier coefficients b_n are all identically equal to \_\_\_\_

Answer 67

The Fourier series of an even function is a "Fourier **_cosine_** series" because the Gourier coefficients b_n​ are all identically equal to **_zero_**

Question 68

The Fourier series of an odd function is a "Fourier ____ series" because the Fourier coefficients a_n, including a₀, are all identically equal to \_\_\_\_

Answer 68

The Fourier series of an odd function is a "Fourier **_sine_** series" because the Fourier coefficients a_n, including a₀, are all identically equal to **_zero_**

Question 69

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

Answer 69

where denotes the fundamental period of x(t)

Question 70

What is Parseval’s Identity for Fourier series?

Answer 70

Question 71

State Parselva's Identity with proof

Answer 71

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