The First Title:
Q1:- Using Differential equations solve the following examples.
Ex1: A circular cylinder of radius 10 m and height 25 m, whose axis is vertical, is filled with water. How long will it take for all the water to escape through an orifice with 50 cm radius at the bottom of the tank? Assuming the velocity of escape v in terms of instantaneous height h is given by 𝒗 = 𝟎. 𝟔√𝟐𝒈𝑯.
Ex2:-an object cools from 100oC to 70oC in 20 min. find the temperature in 40 minutes if the surrounding temperature is 20oc.
Ex3:- A beam of length L is simply supported at both ends as shown in Fig..
(a) Find the deflection if the beam has constant weight W per unit length and (b) determine the maximum deflection.
Ex4:- Find the current in the RLC circuit in Fig.2. Assume I (0) = 0and vL (0) = 0 for R=160 Ω, L=20H, C=0.002 Farads and E(t) = 481cos10t .
The second Title:
Q1:- Using Laplace Transform solve the following examples.
Ex1: A circular cylinder of radius 10 m and height 25 m, whose axis is vertical, is filled with water. How long will it take for all the water to escape through an orifice with 50 cm radius at the bottom of the tank? Assuming the velocity of escape v in terms of instantaneous height h is given by 𝒗 = 𝟎. 𝟔√𝟐𝒈𝑯.
Ex2:-an object cools from 100oC to 70oC in 20 min. find the temperature in 40 minutes if the surrounding temperature is 20oc.
Ex3:- A beam of length L is simply supported at both ends as shown in Fig..
(a) Find the deflection if the beam has constant weight W per unit length and (b) determine the maximum deflection.
Ex4:- Find the current in the RLC circuit in Fig.2. Assume I (0) = 0and vL (0) = 0 for R=160 Ω, L=20H, C=0.002 Farads and E(t) = 481cos10t .
The third Title:
Q1: solve the following examples
Ex1: Example: Find Fourier series for each of the following periodic functions:
1) 𝑓(𝑥) = {0, −𝜋 < 𝑥 < 0 ℎ, 0 < 𝑥 < 𝜋 2) 2) 𝑓(𝑥) = { 𝑥, 0 < 𝑥 < 𝜋
𝜋, 𝜋 < 𝑥 < 2𝜋 Ex2:-Find Fourier transform for:-
1) 𝑓(𝑡) = 𝑒−𝑎𝑡 𝑡 ≥ 0 2) 𝑓(𝑡) = 𝑈(𝑡) |𝑡| < 𝑇 3) 𝑓(𝑡) = 1
(𝑡−1)2+1 𝑢(𝑡 − 1) 4) 𝑓(𝑡) = 𝑒−3|𝑡| cos (4𝑡)
Ex3:-Find inverse Fourier transform for:- 1) 𝑓(𝜔) = 1+3(𝑖𝜔)
(𝑖𝜔)2+5(𝑖𝜔)+6 2) 𝑓(𝜔) = 2−(𝑖𝜔)
(1+𝑖𝜔)(1+𝜔2)
3) 𝑓(𝜔) = sin 𝜔
(𝜔2)
4) 𝑓(𝜔) = 𝑒−𝑖3𝜔
(𝜔2+1)
The fourthTitle:
Q1: solve the following examples
Example1:-Solve the system of linear equations by Gauss - Jordan elimination method and Gauss seidel maehod.
𝒙𝟏 + 𝒙𝟐 + 𝟐𝒙𝟑 = 8 , − 𝒙𝟏− 𝟐𝒙𝟐 + 𝟑𝒙𝟑 = 1 𝟑𝒙𝟏 − 𝟕 𝒙𝟐 + 𝟒𝒙𝟑 = 10
Example2: Solve the system of linear equations by using Gauss Jordon method Gauss seidel maehod.
𝟐𝒙𝟏 − 𝒙𝟐 + 𝟑𝒙𝟑 = −3 , −𝟑 𝒙𝟏 + 𝟐𝒙𝟐 − 𝟔𝒙𝟑 = 7 𝟓𝒙𝟏 − 𝟑 𝒙𝟐 + 𝟖𝒙𝟑 = −9
Example3: Solve the system of linear equations by using Gauss Jordon method and find the inverse of the coefficient matrix Gauss seidel maehod.
𝒙𝟏 − 𝟐𝒙𝟐 + 𝟑𝒙𝟑 = 9 , −𝒙𝟏 + 𝟑𝒙𝟐 = −4 𝟐𝒙𝟏− 𝟓 𝒙𝟐+ 𝟓𝒙𝟑 = 17
The Fifth title
Q1:-
1- Find the straight line that best fits the table:
X 0 1 2 3 4
Y 1 2 3 4 5
2- Using Difference table formulas to find y(1.9), y(2.2), y(6.2), y(10.1)the following data:
(2,-2), (4,1),(6,3),(8,8) and (10,20).
X 2 4 6 8 10
Y -2 1 3 8 20
3- Use Simple iteration method and Newton-Raphson method to estimate the root of 𝑓(𝑥) = 𝑒−𝑥 − 𝑥. Correct up to four decimal places.
Ex4: Evaluate ∫ 𝑒−20 −𝑥2𝑑𝑥 𝑛 = 10 𝑓(𝑥) = 𝑒−𝑥2 using Trapezoidal and simpson rules.
