Systems of ODEs
Firstorder linear equations with constant coefficients
[pic]
[pic]
Let [pic]
[pic]
Taking Laplace transforms of (1) and (2)
[pic]
[pic]
From (3) and (4)
[pic]
[pic]
We solve this system algebraically for [pic]and [pic] and obtain [pic] by taking inverse transforms.
Example [pic]
[pic]
[pic]
We have
[pic]
[pic]
[pic]
From (5) and (6)
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
Since[pic]
[pic]
[pic]
From (II)
[pic]
[pic]
[pic]
[pic]
[pic]
Since[pic]
[pic]
Solution:
[pic]
[pic]
Example
[pic]
Advanced Engineering Mathematics (Kreyszig)
Tank T1 initially contains 100 gal of pure water.
Tank T2 initially contains 100 gal of water in which 150lb of salt are dissolved
The inflow into T1 from outside is 6 gal/min with 6 lb of salt. Obtain the salt content [pic] and [pic] in T1 and T2 respectively. Assume that the mixtures are kept uniform.
The above mechanical system consists of two bodies of mass 1 on 3 springs of the same spring constant k. The masses of the springs are negligible. When the system is in static equilibrium, the masses m1 and m2 are at positions A and B respectively, and the springs all have extension zero as the masses of m1 and m2 are small compared with the spring constant. Each spring is neither extended nor compressed.
With the masses in the positions C and D as shown (above right),
Extension of top spring = [pic]
Extension of middle spring = DC – AB
= (PD – PC) – AB
= [(l1 + l2 + y2) – (l1 + y1)] – l2
= y2 – y1
Consider mass x acc. = Force(mass = 1)
For a stretched spring F =  k x (where k is the spring constant and x is the extension)
...Solving systems of linearequations
7.1 Introduction
Let a system of linearequations of the following form:
a11 x1
a21 x1
a12 x2
a22 x2
ai1x1 ai 2 x2
am1 x1 am2 x2
a1n xn
a2 n x n
ain xn
amn xn
b1
b2
bi
bm
(7.1)
be considered, where x1 , x2 , ... , xn are the unknowns, elements aik (i = 1, 2, ..., m;
k = 1, 2, ..., n) are the coefficients, bi (i = 1, 2, ..., m) are the free terms of the system. In
matrix notation, this system has the form:
Ax b ,
(7.2)
where A is the matrix of coefficients of the system (the main matrix), A = [aik]mn, b is the
column vector of the free terms, bT [b1 , b2 , ... , bm ] , x is the column vector of the
unknowns, xT [ x1 , x2 , ... , xn ] ; the symbol () T denotes transposition.
It is assumed that aik and bi are known numbers. An ordered set, {x1, x2, ..., xn}, of real
numbers satisfying (7.1) is referred to as the solution of the system, and the individual
numbers, x1, x2, ..., xn, are roots of the system.
A system of linearequations is:
consistent  if it has at least one solution. At the same time it can be

determined  if it has exactly one, unique solution,...
...reason other than lowering a high school student's grade point average. Systems of linearequations, or a set of equations with two or more variables, are an essential part of finding solutions with only limited information, which happens to be exactly what algebra is. As a required part of any algebra student's life, it is best to understand how they work, not only so an acceptable grade is received, but also so one day thesystems can be used to actually find desired information with ease.
There are three main methods of defining a system of linearequations. One way is called a consistent, independent solution. This essentially means that the system has one unique, definite solution. In this situation on a graph, a set of two equations and two variables would be solved as one single point where two lines intersect. It is much the same with three variables and three equations. The only difference is that the point is an intersection of three planes instead of two lines.
Additionally, there are situations where a system of linearequations could be described as consistent, dependent. These systems of linearequations have an infinite number of solutions where a general solution is used to substitute one or two variables for one other...
...SYSTEM OF LINEAREQUATIONS IN TWO VARIABLES Solve the following systems: 1.
x y 8 x y 2
by graphing
by substitution
by elimination
by Cramer’s rule
2.
2 x 5 y 9 0 x 3y 1 0
by graphing
by substitution
by elimination
by Cramer’s rule
3.
4 x 5 y 7 0 2 x 3 y 11 0
by graphing
by substitution
by elimination
by Cramer’s rule
CASE 1: intersecting lines independent & consistent m1m2
CASE 2: parallel lines inconsistent m1 = m2 ; b1 b2
CASE 3: coinciding lines consistent & dependent m1 = m2 ; b1 = b2
Classify the following system, whether (a) intersecting, (b) parallel, or (c) coinciding lines 1.
3 x 4 y 1 0 3 x 4 y 2 0 3 x 4 y 1 0 6 x 8 y 2 0
Solve the following systems in three variables: 1. 3 x 4 y z 1 2. x y 2
x 4 y 3z 3 3 x 2 y 2 z 0
________
3 y z 1 x 2 z 7
2.
________
3.
2 x 5 y 1 0 5 x 2 y 2 0
2 x y 1 4 x 2 y 3 x 2 y 1 0 2 x y 1
________
4.
________
5.
________
1 x Solve 1 x
2 3 y 3 2 y
Problem solving Form a system of equations from the problems given below. A) (MIXTURE PROBLEM 1) How many pounds of a 35% salt solution and a 14% salt solution should be combined so that a 50...
...Patterns within systems of LinearEquations
HL Type 1 Maths Coursework
Maryam Allana
12 Brook
The aim of my report is to discover and examine the patterns found within the constants of the linearequations supplied. After acquiring the patterns I will solve the equations and graph the solutions to establish my analysis. Said analysis will further be reiterated through the creation of numerous similar systems, with certain patterns, which will aid in finding a conjecture. The hypothesis will be proven through the use of a common formula. (This outline will be used to solve both, Part A and B of the coursework)
Part A:
Equation 1: x+2y= 3
Equation 2: 2xy=4
Equation 1 consists of three constants; 1, 2 and 3. These constants follow an arithmetic progression with the first term as well as the common difference both equaling to one. Another pattern present within Equation 1 is the linear formation. This can be seen as the equation is able to transformed into the formula ‘y = mx+c’ as it is able to form a straight line equation (shown below). Similar to Equation 1, Equation 2 also follows an arithmetic progression with constants of; 2, 1 and 4. It consists of a...
...2014/9/16
LinearEquations
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LinearEquations
A linearequation is an equation for a straight line
These are all linearequations:
y = 2x+1
5x = 6+3y
y/2 = 3 x
Let us look more closely at one example:
Example: y = 2x+1 is a linearequation:
The graph of y = 2x+1 is a straight line
When x increases, y increases twice as fast, hence 2x
When x is 0, y is already 1. Hence +1 is also needed
So: y = 2x + 1
Here are some example values:
http://www.mathsisfun.com/algebra/linearequations.html
x
y = 2x + 1
1
y = 2 × (1) + 1 = 1
0
y = 2 × 0 + 1 = 1
1
y = 2 × 1 + 1 = 3
1/6
2014/9/16
LinearEquations
2
y = 2 × 2 + 1 = 5
Check for yourself that those points are part of the line above!
Different Forms
There are many ways of writing linearequations, but they usually have constants (like "2" or
"c") and must have simple variables (like "x" or "y").
Examples: These are linearequations:
y = 3x 6
y 2 = 3(x + 1)
y + 2x 2 = 0
5x = 6
y/2 = 3
But the variables (like "x" or "y") in LinearEquations do NOT have:
Exponents (like the 2 in x2)
Square roots, cube roots, etc...
...Summer 20103 CLASS NOTES CHAPTER 1
Section 1.1: LinearEquations
Learning Objectives:
1. Solve a linearequation
2. Solve equations that lead to linearequations
3. Solve applied problems involving linearequations
Examples:
1. [pic]
[pic]
3. A total of $51,000 is to be invested, some in bonds and some in certificates of deposit (CDs). If the amount invested in bonds is to exceed that in CDs by $3,000, how much will be invested in each type of investment?
4. Shannon, who is paid timeandahalf for hours worked in excess of 40 hours, had gross weekly wages of $608 for 56 hours worked. What is her regular hourly wage?
Answers: 1. [pic]
2. [pic]
3. $24,000 in CDs, $27,000 in bonds 4. $9.50/hour
Section 1.2: Quadratic Equations
Learning Objectives:
1. Solve a quadratic equation by (a) factoring, (b) completing the square, (c) the
quadratic formula
2. Solve applied problems involving quadratic equations
Examples:
1. Find the real solutions by factoring: [pic]
2. Find the real solutions by using the square root method: [pic]
3. Find the real solutions by completing the square: [pic]
4. Find the real solutions by using the quadratic formula: [pic]
5. A...
...for Weeks One and Two
Chapter 4 Systems of LinearEquations; Matrices (Section 41 to 46)  Examples  Reference (Where is it in the text?) 
  
DEFINITION: Systems of Two LinearEquations in Two VariablesGiven the linearsystem ax + by = hcx + dy = kwhere a , b , c , d , h , and k are real constants, a pair of numbers x = x0 and y = y0 [also written as an ordered pair (x0, y0)] is a solution of this system if each equation is satisfied by the pair. The set of all such ordered pairs is called the solution set for the system. To solve a system is to find its solution set.  EXAMPLE 1 Solving a System by Graphing Solve the ticket problem by graphing2x + y = 8x + 3 y = 9SOLUTION An easy way to find two distinct points on the first line is to find the x and y intercepts.Substitute y = 0 to find the x intercept (2x = 8, so x = 4), and substitute x =0 to find the y intercept (y = 8).Then draw the line through (4, 0) and (0, 8).After graphing both lines in the same coordinate system (Fig. 1), estimate the coordinates of the intersection point:  Page 170 
Systems of LinearEquations: Basic TermsDefinition: A system of linearequations is consistent if...
...TR 3923 Programming Design in Solving Biology Problems Semester 1, 2011/2012
Elankovan Sundararajan School of Information Technology Faculty of Information Science and Technology
TR 3923 Elankovan Sundararajan 1
Lecture 3
System of LinearEquations
TR 3923
Elankovan Sundararajan
2
Introduction
• Solving sets of linearequations is the most frequently used numerical procedure when realworld situations are modeled. modeled Linearequations are the basis for mathematical models of
1. 2. 2 3. 4. 5. Economics, Computational Biology Comp tational Biolog and Bioinformatics Bioinformatics, Weather prediction, Heat and mass transfer, Statistical analysis, and a myriad of other application.
•
•
TR 3923
The methods for solving ODEs and PDEs also depend on them.
Elankovan Sundararajan
3
System of LinearEquations
• Consider the following general set of n equations in n unknowns: a11 x1 a12 x2 a1n xn c1 : R1
a21 x1 a22 x2 a2 n xn c2 , an 1,1 x1 an 1, 2 x2 an 1,n xn cn 1 , an ,1 x1 an , 2 x2 an ,n xn cn .
Which can be written in matrix form as:
: R2
: R n1 : Rn
A x b.
~ ~
TR 3923 Elankovan Sundararajan 4
where, A is the nxn matrix, ,
a11 a12 a1n a21 a22 a2 n A . a an 2...
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