TABLE OF CONTENT
CHAPTER ONE
1.0 INTRODUCTION
1.1 DEFINITION OF TERMS
1.2 SOLUTIONS OF LINEAR EQUATIONS
CHAPTER TWO
2.0 SIMULTAENOUS LINEAR DIFFERENTIAL EQUATION WITH CONSTRAINTS COEFFICIENTS.
2.1 LINEAR OPERATOR
CHAPTER THREE
3.0 APPLICATION OF SIMULTAENOUS DIFFERENTIAL EQUATIONS AND EXAMPLES
3.2 APPLICATION TO ASTRONOMY
CHAPTER FOUR
4.0 SUMMARY
4.1 CONCLUSION
4.2 RECOMMENDATION
REFERENCES
CHAPTER 1
1.0 INTRODUCTION
In mathematics, the history of differential equation traces the development of “D.E” from calculus, with itself was independently inverted by English physicist Isaac Newton and German mathematician Gottfried Lebnize.
Sir Isaac Newton (1642-1727) and Goltfied wihelm Leibniz (1646-1716) were the two prominent scientists who independently discovered the fundamental ideas of calculus. This provides a tool for solving problems involving motion and other physical phenomena such as elacticity, analysis of the bending of beams and the shape a string will form under various conditions.
Generally, first-order and higher-order differential equations problems analytically.
Langrange said of Euler’s work in mechanics identified the condition for exactness of first order differential equation in (1734-1735) developed the theory of integrating factors and gave the general solution of homogeneous.
The brothers jakob (1654-1705) and johann (1667-1748) bernailli of basel did much to develop methods of solving differential equations, with the aid of calculus they formulated as differential equations and solved a number of problems in mechanics. In the same paper (in 1690) he firt used the term “integral” in the modern sense. In 1694 johann Bernoulli was able to solve the equation
Even though it was not yet known that d (In x) =
Which led to much friction between them, was the brachristochrone problem. The determination of the curve of fastest decent.
The brachistrochone was also solved by Leibniz and newton in addition to Bernoulli brother.
The study of differential equation originated in the beginnings of the calculus. Newton did relatively work in differential equations, his development of the calculus and elucidation of the basic principles of mechanics provided a basis for their application in the eighteenth century, most notably by Euler. Newton classified first order differential equation which is written in the form = f(x)
= f (y) and = f(x,y)
By the end of the eighteenth century many elementary methods of solving ordinary differential equations had been discovered. In the nineteenth century interest turned toward the investigation of the theoretical equations of existence and uniqueness and to the development of less elementary methods such as those based on power series method. Partial differential equations also began to be studied intensively, as their crucial role in mathematical physics became clear.
In 1693, Leibniz solved his first differential equation and that same year newton published the results of previous D.E solution methods a year that is said to mark the inception for D.E as a distinct field in mathematics.
1.1 DEFINATION OF TERMS
This section focuses mainly on the basis terminologies associated with the study of first-order ordinary differential equations.
A differential equation is an equation that contains the derivatives or differentials of one or more dependent variables, with respect to one or more independent variables.
Types of differential equations
- An ordinary differential equation
- A partial differential equation
- An ordinary differential equation is an equation which contains only ordinary derivatives of one or more independent variables, with respect to a single independent variables.
For example
– y = 2
-
Are ordinary differential equations
Ii a partial differential equation is an equation is an equation involving the partial derivatives of one or more dependent variables with respect to two or more independent variables.
For example,
Are partial differential equations
Order of differential equations is the order of the highest derivative in a differential equation. For example
+ (
Is a second order differential equation
+ y = 0
Is a first order differential equation
K2
Is a third-order partial differential equation.
Degree of Differential Equation;- is the degree of the highest order of the differential equation. For example
(
Is of degree 3
K2 is of degree 1
Linear and non-linear differential equations
A differential equations is said to be linear if it has the form
It should be observed that linear differential equation are characterized by two properties.
- The dependent variable y and its derivatives are of the first degree; and
- Each coefficient depends only on the independent variable x. an equation that is not linear is said to be non-linear.
the equation x dy + y dx = 0
y’’ – y’ + y = 0 and – y =
Are linear first, second and third order differential equations, respectively on the other hand
And
are non-linear first, second and third
Order ordinary differential equations respectively.
Linear equation;- we defined the general form of linear D.E of order ‘n’ to be
The coefficients are functions of x only, and that y all its derivatives are raised to the first power, now if n=1 we obtain the linear first order equation
Dividing all by a1(x), we have
-------------------------------------------(1)
Where P(x) and Q(x) are continuous function. It follows that (1) has the following solution. Equation (1) can be written as
-dy + [p (x) y – Q(x) ]dx = 0-----------------------------------------(2)
Linear equation posses the pleasant property that x function u(x) can always be found such that, the multiple of (2)
……………………………………..………(3)
In an exact D.E using exact
………………..………………………….……………..(4)
=
Using variable seperable, to determine u(x)
In / / =
(x) = --------------------------------------------(5)
The function u(x) defined in (5) is called integrating factor for the linear equation.
From equation (4), we solve
+
+
Are exact differentials we now write (3) in the form
+
=
Integrating both sides, gives
=
Y = ---------------------------------------------------(6)
The equation (1) has a solution it must be of from (6)
VARIABLE SEPARABLE;- if g(x) is a given continuous function then the first –order equation
--------------------------------------------------------------------------(1)
Can be solve by integrating. the solution (1) is
Y =
OR
A D.E of the form
Is said to be separable
hy ……………………………………………………….(2)
Now if y = f(x) denotes a solution of (2) we now have
h[f(x)) f’(x) = g(x) therefore
…………………………………..(3)
But dy =f1 (x)dx so (3), becomes
…………………………………………………......(4)
Equation (4) is the required solution.
Initial value and boundary- value problems: for a linear nth-order differential equation the problem solve
………………..(1)
Subject to :
Where y0, y01---------------y0(n-1) are arbitrary constant, is called an initial-value problem. we seek a solution on some interval I containing the point x=x0
In the important case of a linear second-order equation, a solution of
Y(x0) = yo
Y’(xo) = yo’
Is a function defined on I whose graph passes through (X0,yo) such that the slope of the curve at the point is the number Y10.
The next theorem give sufficient conditions for the existence of a unique solution (1)
THEOREM 4.1
Let an(x), an-1(x)--------, a1(x), a0(x) and g(x) be continuous on an interval I and let an(x)0 for every x in this interval. If x=x0 is any point in this interval, then a solution y(x) of the initial-value problem (1) exists on the interval and is unique.
While we are not in a position to prove theorem 4.1 in its full generality a demonstration of the uniqueness of the solution in the special case.
A2y’’ + a1y’ + a0y = g(x)
Y(0) = y0
Y’(0) = y’0
Where a2,a1and a0 are positive contants and g(x) is continuous for all x.
The requiremets in the theorem 4.1 that ai(x), i=1,2…n be continuous and an(x) 0 for every x in I are both important specifically, if an(x)=0 for some x in the interval then the solution of linear initial –value problem may not be unique or even exist.
Another type of problem consists of solving differential equation of order two or greater in which the dependent variable y(or its derivatives) is specified at two different points
A problem such as
Subject to : y(a)=y0, y(b) = y, is called a two-point boundary-value problem or simply a boundary-value problem.
4.1.2 linear dependence and linear independence: the next two concepts are basic to the study of linear differential equations.
DEFINATION: 4.1 a set of function F1(x), F2(x),……, Fn(x) is said to be linearly dependent on an interval I if there exist constatnts C1, C2, ……, Cn , not all zero, such that
C1 F1 (x) + C1, F2(x) + ------ +CnFn(x) =0
For every x in the interval.
DEFINATION 4.2 A set of function F1(x), F2 (x),…. Fn (x) is used to be linearly independent on an interval I if it is not linearly dependent on the interval.
In other words, a set of functions is linearly independent on an interval if the only constant for which
C1 F1 (x) + C1, F2(x) + ------ CnFn(x) =0
For every x in the interval, are C1 = C2 = ---- = Cn = 0.
It is easy to understand these definitions in the case of two functions F1(x) and F2(x). if the functions are linearly dependent on an interval then there exist constant C1 and C2
Which are not both zero such that for every X in the interval
C1 F1 (x) + C2F2(x) =0
Therefore, if we assume that C1 0, it follows that F1 (x) = F2(x) that is, if two fucntions are linearly dependent then one is simply a constant multiple of the other. Conversely, if F1(x) = C1F2(x), for some constant C2, then
(-1). F1(x) + C2 F2 (x) =0
For every X on some interval. Hence the functions are linear dependent since at least one of the constants (namely, C1=-1) is not zero. We conclude that two functions are linear independent when neither is a constant multiple of the other on an interval.
THEOREM 4.2
Suppose F1(x), F2(x), …………. Fn(x) posses at least n-1 derivatives. If
For at least one point in the interval I. then, the function F1(x), F2(x),…….. ,Fn(x) are linearly independent on the interval.
The determinant in the above theorem is denoted by
W (f1 (x), f2 (x), ------, fn(x)) and is called the Wronskian of the functions.
PROOF: we prove 4.2 by contradiction for the case when n=2. Assume that
W(F1(x0), F2(x0) ) 0 for a fixed x0 in the interval I and that F1(x) and F2(x) are linearly dependent on the interval. The fact that the function are linearly dependent means there exist constant C1 and C2 not both zero for which C1F1 (x) + C2 F2 (x) =0 for every x in I. differentiating this combination then gives
C1 F’’1 (x) + C1 F’’2 (x) =0
Thus we obtain the system of linear equation
But the linear dependence of F1 and F2 , implies that (2) possesses a non trivial solution for each x in the interval. Hence
W (F1 (x), F2 (x) ) = = 0
For every x in I. this contradicts the assumption that W(F1 (xo), F2 (xo)) 0. We conclude that F1 and F2 are linearly independent.
COROLLARY if F1 (x), F2 (x),…. Fn (x) possess at least n-1 derivative and are linear dependent on I then the interval that is W(F1 (x), F2 (x),…….., Fn(x)) 0 on the interval.
4.1.3 SOLUTIONS OF LINEAR EQUATIONS: HOMOGENOUS EQUATIONS;-
A linear nth – order differential equation of the form
……………………………..(3)
Is said to be homogenous where as
…………………….(4)
g(x) not identically zero, is said to be non-homogenous.
Note: to avoid needles repetition throughout the remainder of this text we shall as a matter of course make the following important assumptions when given definitions and proving theorems about the linear equation (3) and (4) on some common interval I.
(1) The coefficient ai (x), i=0,1,…, n are continuos
(2) The right-hand member g(x) is continuous
(3) And an(x)0 for every x in the interval.
The following theorem is known as the superposition principle
THEOREM 4.3
Let y1, y2 ………….., yk be solutions of the homogenous linear nth-order differential equation (3) on an interval I. then the linear combination
Y = C1Y1(x) + C2Y2(x) + ----------------+ CkYk (x)-----------------------------------(5)
Where the Ci, I = 1, 2, -----k are arbitrary constants, is also a solution on the interval.
Proof:
We prove the case when n=k =2. Let y1(x) and y2(x) be solution of
A2(x) y’’ + a1 (x) y’ + a0 (x) y =0 if we defined
Y = C1Y1(x) + C2Y2 (x) then
A2(x) [C1Y’’1 + C2Y’’2 ] + a1(x) [C1Y’1 + C2Y’1] + a0(x) [C1Y1 + C2Y2]
C1 [a2 (x) y’’1 + a1(x) Y’1 + ao (x) y1] + C2 [a2 (x) y’’2 + a1(x) Y’’2 + a0 (x) Y2 ]
= C1.0 + C2.0
=0
DEFINATION
Let yp be a given solution of the homogenous linear nth-order differential equation (4) on an interval I and let
Yc = c1y1 (x) + c2y2 (x) +-----------------+Cnyn (x)
Denote the general solution of the associated homogenous (3) on the interval. The general solution of the non-homogenous equation, on the interval is defined to be
y= c1y1 (x) + c2y2 (x) +-----------------+Cnyn (x) + Yp (x) = yc + yp
From the above definition, the general solution of (3), is called the complementary function for equation (4). In other words, the general solution of a form non homogenous linear differential equation is
y= complementary function + any particular solution.
Homogenous Linear Equation with constant coefficient:-
We have seen that the linear first-order equations + ay = 0, where a is a constant, has the experimental solution y= C1e-ax on -∞ < x < ∞. Therefore it is natural to seek to determine whether exponential solutions exist on -∞ < x < ∞ for higher order equation such as
-----------------------------(1)
Where ai, I = 0, 1, 2 ----, n are constants. The solutions of (1) are exponential function or constructed out of exponential functions. We shall begin by considering the special case of the second order equation.
ay’’ + by’ + cy = 0----------------------------------------------------------------------------------- (2)
If we try a solution of the form y=emx , then y’= memx and y’’= m2emx so that equation (2) becomes am2mx + bmmx + cmmx = 0
Or
mx (am2 + bm + c) = 0
Because mx is never zero for real value of x, it is apparent that the only way that this exponential function can satisfy the differential equation is to choose in so that it is root of the quadratic equation
am2 + bm + c = 0--------------------------------------(3)
Where equation (3) is called the auxillary equation or characteristics equation of the differential equation (2). We shall consider three cases namely, the solutions of the auxillary equations corresponding to real distinct roots, real but equal roots, and complex conjugate roots.
CASE 1. The auxillary equation (3) has two unequal roots, M1 and M2 we find two solution y1=m1xand y2= m2xsince these functions are linearly independent on -∞ < x < ∞
It follows the general solution of (2) on this interval is
Y = c1em1x + c2em2x------------------------------------------(4)
CASE II when m1= m2 we obtain only one exponential solution Y1=c1em1x
However, the second solution is
Y2=em1x---------------------------------------(5)
But from quadratic formular we have
M = -b
Since m1=m2 to have b2-4ac = 0. Implies
2m1 = …………………………..(5) becomes
Y2 = em1x = em1x
The general solution of (2) is then
Y=C1em1x + C2xem1x………………………..(6)
CASE III: if M1 & M2 are complex then we can write m1=α + βi and M2 = α - βi where α and βare real and i2 =-1. Where
Y=c1 c2 -----------------------------------------(7)
Using Euler’s formular
where is any real number from this result
We can write
Thus (7) becomes
Y = [c1 c2
=
=
Since and themselves form a fundamental set of solutions of the given differential equation on -∞ < x < ∞ we can
