ORDINARY DIFFERENTIAL EQUATIONS PREPARED BY RAIFU S. AJAYI NATIONAL UNIVERSITY OF RWANDA BUTARE 2011

TABLE OF CONTENTS :

TABLE OF CONTENTS Basic Concepts: Definitions, Direction Fields First Order Differential Equations: Linear. Separable, Exact Equations, Bernoulli Differential Equations, Substitution Methods, Modeling With First Order Differential Equations, Equilibrium Solutions and Euler’s Method. Second Order Differential Equations: Basic Concepts, Real Roots, Complex roots, reduction of orders, fundamental sets of solutions, Wronskian , non-homogeneous differential Equations, method of undetermined coefficients, variation of parameter methods, mechanical vibrations.

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Laplace Transforms: Definition, Laplace transforms, Inverse Laplace Transforms, Step, Heavy-Side and Delta-Dirac Function, Solving INP with Laplace Transforms, Nonconstant Coefficient IVP’s Convolution Integral, Table of Laplace Transforms. System of Differential Equations: solution Methods, Real eignevalues , complex and repeated eigenvalues , non-homogeneous systems, Laplace transform, Modeling. Power Series Solutions Higher Order Differential Equations. Boundary value problem and Fourier Series Special Differential Equations Introduction to Numerical Methods of Solving ordinary Differential Equations.

DIFFERENTIAL EQUATION IN ACTIONS:

DIFFERENTIAL EQUATION IN ACTIONS

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True brevity is the ability to stand firm to do the right thing, even when the people and circumstances seem to appeal against it.

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Visualization of heat transfer in a pump casing, by solving the heat equation . Heat is being generated internally in the casing and being cooled at the boundary, providing a steady state temperature distribution.

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Newton's third law. The skaters' forces on each other are equal in magnitude, but act in opposite directions

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The analysis of projectile motion is a part of classical mechanics

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Each turn of the cell cycle divides the chromosomes in a cell nucleus

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Biston betularia f. typica is the white-bodied form of the peppered moth

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Biston betularia f. carbonaria is the black-bodied form of the peppered moth.

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Staphylococcus aureus (colloquially known as "Staph aureus" or a Staph infection ) is one of the major resistant pathogens. Found on the mucous membranes and the human skin of around a third of the population

This figure illustrates the relative abilities of three different types of ionizing radiation to penetrate solid matter. Alpha particles (α) are stopped by a sheet of paper whilst beta particles (β) halt to an aluminium plate. Gamma radiation (γ) is dampened when it penetrates matter:

This figure illustrates the relative abilities of three different types of ionizing radiation to penetrate solid matter. Alpha particles (α) are stopped by a sheet of paper whilst beta particles (β) halt to an aluminium plate. Gamma radiation (γ) is dampened when it penetrates matter

Center of Momentum Frame, the decay of a particle into two equal mass particles results in them being emitted with an angle of 180° between them. ...while in the Lab Frame the parent particle is probably moving at a speed close to the speed of light so the two emitted particles would come out at angles different than that of in the center of momentum frame. :

Center of Momentum Frame , the decay of a particle into two equal mass particles results in them being emitted with an angle of 180° between them. ...while in the Lab Frame the parent particle is probably moving at a speed close to the speed of light so the two emitted particles would come out at angles different than that of in the center of momentum frame.

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The making of atomic bumb or nuclear bumb depends On mathematics.

circuit:

circuit

Suspension bridge cable :

Suspension bridge cable

The Apollo 15 capsule landed safely despite a parachute failure. (velocity) :

The Apollo 15 capsule landed safely despite a parachute failure. (velocity)

Electromagnetism :

Electromagnetism

A number of modes are shown below together with their quantum numbers. The analogous wave functions of the hydrogen atom are also indicated. :

A number of modes are shown below together with their quantum numbers. The analogous wave functions of the hydrogen atom are also indicated. Mode u 01 (1s) Mode u 02 (2s) Mode u 03 (3s) Mode u 11 (2p) Mode u 12 (3p) Mode u 13 (4p) Mode u 21 (3d) Mode u 22 (4d) Mode u 23 (5d)

Vibrating string:

Vibrating string

Annotated color version of the original 1824 Carnot heat engine showing the hot body (boiler), working body (system, steam), and cold body (water), the letters labeled according to the stopping points in Carnot cycle:

Annotated color version of the original 1824 Carnot heat engine showing the hot body (boiler), working body (system, steam), and cold body (water), the letters labeled according to the stopping points in Carnot cycle

...while in the Lab Frame the parent particle is probably moving at a speed close to the speed of light so the two emitted particles would come out at angles different than that of in the center of momentum frame. :

... while in the Lab Frame the parent particle is probably moving at a speed close to the speed of light so the two emitted particles would come out at angles different than that of in the center of momentum frame.

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DEFINITION: A differential equation is any equation which contains derivatives, either with one independent variable or more than one independent variables. A differential equation is ordinary differential equation if it involves one ore several derivatives of an unspecified function y of x. Partial Differential Equation: This is a differential equation that contains more than one independent varibales Fractional Differential Equation: This is a differential Equation of arbitrary order. Order: The order of a differential equation is the largest derivatives present in the differential equation. Thus a first order will contain and may also contains y and function of x.

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A linear differential equation of order n is a differential equation written in the following form: where is not the zero function. Note that some may use the notation for the derivatives. System of Differential Equations : The vector given as

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is called a first order system (of differential equations). It represents the system (or set) of first order differential equations. System of differential equations have a natural interpretation as a description of a dynamical system.

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A differential equation is a description of a dynamical system. Differential Equation are also classified as Linear or Non-linear. A differential equation is linear if it depends linearly on y, regardless of the dependence on x. Any other form of differential equation which is not linear is called non-linear differential equation

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The characteristic property of linear equations is that their solutions form an affine subspace of an appropriate function space, which results in much more developed theory of linear differential equations. Homogerneous Linear Differential Equation: This is a subclass of linear equation for which the space of solutions is a linear subspace i.e the sum of any set of solutions or multiples of solutions is also a solution A linear differential equation has f(x) = 0 in the above equation is called homogeneous linear equation, Otherwise it is called non-homogeneous.

Example of differential equations:

Example of differential equations

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Differential Operator: The operator D is a shorthand notation to express differential equation in a simple ways or short form such as. t Differential Equation can be Delay Differential Equation: This is an equation of a single variable, usually called time, in which the derivative of the function at a certain time is given in terms of the values of the function at earlier times

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Stochastic Differential Equation: This is an equation in which the Unknown quantity is a stochastic process and the equation involves some known stochastic processes e.g, the Weiner process in diffusion equations. NOTABLE DIFFERENTIAL EQUATIONS Newton’s Second Law in dynamics (mechanics) The law states that the net force on a particle is equal to the time rate of change of its Linear momentum p in an inertial reference frame. Hamilton’s equations in classical mechanics: The value of the Hamiltonian is the total energy of the system being described. It is the sum of the kinetic and potential energy in the system for a closed system. It can be used to describe systems such as bouncing ball, a pendulum, or oscillating spring, planetary orbits in celestial Mechanics. Generally express as

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Radioactive decay in nuclear physics: This is the process by which an unstable atomic nucleus loses energy by emitting ionizing particles or Radiation. In equation form we have; The wave equations: This is a PDE of second order, such as sound waves, light waves and water waves. It arises in acoustic, electromagnetic and fluid dynamics. It is of the form Newton’s law of cooling in thermodynamics: This states that the rate Of Heat loss of a body is proportional to the difference in temperatures Between the body and its surroundings. In equation form, we have

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MAXWELL EQUATIONS: This are set of four partial differential equations describing how the electric and magnetic fields relate to their sources, charge density and current density, and how they develop with time.

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Formulation in terms of free charge and current Name Differential Form Integration Gauss’s Law Gauss’s Law for Magnetism Maxwell-Faraday equation (Faradays law of induction) Ampere’s circuit law (with Maxwell correction

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Formulation in terms of total charge and current Name Differential Form Integration Gauss’s Law Gauss’s Law for Magnetism Maxwell-Faraday equation (Faradays law of induction) Ampere’s circuit law (with Maxwell correction

Heat Equation :

Heat Equation The heat equation in PDE describes the distribution of heat (or variation in temperature) in a given region over time. It is of the form:

LAPLACE EQUATION:

LAPLACE EQUATION Laplace equation is very important in general theory of solution of potential theory. The solution are all harmonic and can be applied in the field such as electromagnetism, astronomy and fluid dynamics. It is of the form: where is Laplace operator and u is a scalar function. POISSON’S EQUATION : This is non-homogeneous of Laplace equation and it has broad utility in electrostatics, mechanical engineering and theoretical physics. It is of the form:

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Laplace’s Equation on an annulus (r = 2 and r = 4) with Dirichlet Boundary Conditions u(r = 2) = 0 and u(r = 4) = 4sin4

LOGISTIC FUNCTION (VERHULST EQUATION):

LOGISTIC FUNCTION (VERHULST EQUATION) This equation is used to study population growth. The logistic function find applications in a range of fields, including artificial neural network biology, biomathematics, demography, economics, chemistry, mathematical psychology, probability, sociology, political science and statistics. The equation is of the form: THE INDIVIDUAL GROWTH MODEL The model is widely used in biological models and exists in a number of permutations and simply called Von Bertalanffy growth equation with length L over time t.

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LOTKA-VOLTERRA EQUATION: This equation is called predator-prey equations, are a pair of first-order non-liner, differential equations frequently used to describe the dynamics of biological system in which two species interact, one a predator and one its prey. The equation is of the form:

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REPLICATOR EQUATION: This is deterministic monotone non- innovative game dynamic used in evolutionary game theory, it is of the form

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BLACK SCHOLES MODEL: This is a mathematical description of financial markets and derivatives investment instruments. The equation is of the form

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Maltusian Growth Model: This is simply exponential growth model. Is essentially based on a constant rate of compound interest in population dynamics. The model is define as: ADVERTISING MODEL: The Sethi advertising model provides a sales-advertising dynamics in the form of Stochastic equations as:

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SCHRODINGER EQUATION: This is an equation in quantum mechanics that describe how quantum state of a physical system changes in time. The equation is of the form

Navier stokes equations:

Navier stokes equations These equations arise from applying Newton’s second law to fluid motion, together with the assumption that the fluid stress is the sum of a diffusing Viscous term (proportional to the gradient of velocity), plus a pressure Term define as

Cauchy-riemann equations:

Cauchy- riemann equations This equation arises in complex analysis and consist of a system of two partial differential equations that provides a necessary and sufficient condition for a differentiable function to be holomorphic in an open set. The C-R equations on a pair of real-valued function u(x, y) and v(x, y) where u is the real part and v is the imaginary part of the complex function f(z) define as

The concepts of solution:

The concepts of solution

Example :

Example Verify that the given function is the general solution of the differential equation given.

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Verify if the given function satisfy the differential equation given. INITIAL-VALUE PROBLEM An initial value problem is a differential equation combined with initial condition.

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Example 1: Solve the equation

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Example 2: Solve the equation

METHOD OF SOLVING FIRST ORDER DIFFERENTIAL EQUATIONS:

METHOD OF SOLVING FIRST ORDER DIFFERENTIAL EQUATIONS SEPARABLE DIFFERENTIAL EQUATIONS A separable differential equation is an equation of the form

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THEOREM: The differential equation is separable if and only if

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REDUCTION TO VARIABLE SEPARABLE If the problem is implicit and of the form Then integrate to get the final solution.

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Example 1. Find the solution to the problem

transformation:

transformation CASE I: If the differential equation in question is not homogeneous but is of the form below, we use the suggested transformation and solve the problem by separating the variable

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EXAMPLE 3: Solve the equation

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CASE II: If the differential equation is not homogeneous and f(x, y) is a linear function as define below, then we solve as suggested

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EXACT DIFFERENTIAL EQUATION A differential equation of the form:

Example 2:

Example 2

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EXAMPLE 3 Solve the initial value probem

Integrating factors:

Integrating factors If we are faced with a differential equation that is neither of the above rules, and not exact, then we seek for solving such a problem using the procedure below

THEOREM:

THEOREM K(x, y) z X y xy

Example 1:

Example 1 Solve the equation

First order linear non-homogeneous differential equation:

First order linear non-homogeneous differential equation

Example 1:

Example 1 Solve the equation

example 2 :

example 2 Solve the equation

Jacob bernoulli (1654-1705) swiss mathematician:

Jacob bernoulli (1654-1705) swiss mathematician

Solving bernoulli equations:

Solving bernoulli equations

Example 1:

Example 1

EXAMPLE 2:

EXAMPLE 2

Riccati equation :

Riccati equation Jacopo F Riccati (1676-1754)

Reduction to second order:

Reduction to second order

Second order linear equation with constants coefficients:

Second order linear equation with constants coefficients

Solution algorithm:

Solution algorithm

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Niels Henrik Abel 1802-1827 (Norway) Józef Maria Hoene-Wroński 1776-1853 (Poland)

LAPLACE TRANSFORM METHOD The Laplace transform is an integral transform used in solving physical problems particularly those that arises in the analysis of electronics circuits, harmonic oscillators, optical devices and mechanical system. In this analysis the transform is interpreted as a transformation fromtime-domain in which the input and output are functions of time to the frequency domain in which the inputs and outputs are functions of complex angular frequency, in radian per unit time.

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PIERRE SIMON LAPLACE 1749-1827 FRANCE PIECEWISE FUNCTION A function is piecewise continuous on an interval If the interval can be broken into a finite number of subintervals on which the function is continuous

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on each open subinterval and has a finite limit at the end points of each subinterval. Piecewise Continuous Function DEFINITION: Suppose that f(t) is a continuous function. The Laplace transform of f(t) is denoted and define for all real number as:

THEOREM:

THEOREM

Table of laplace transform:

Table of laplace transform N F(s) f ( t ) , t > 0 1.1 definition of a transform y(t) 1.2 Y(s) inversion formula 1.3 sY (s) - y(0 ) first derivative y ' (t) 1.4 second derivative y " (t) 1.5 nth derivative 1.6 (1/s) F(s) integration 1.7 F(s)G(s) convolution integral 1.8 f ( a t) 1.9 F(s - a ) shifting in the s-plane exp (- a t) f(t) 1.10 f(t) has period T, such that f( t + T ) = f (t) 1.11 g(t) has period T, such that g(t + T ) = - g(t)

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Function F(s) F(t) Ideal delay Unit impulse 1 Error Function Delayed nth power frequency shift nth power (for integer n) qth power (for complex q) unit step u(t)

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Function F(s) F(t) delay unit function ramp nth power with frequency shift Exponential decay exponential approach sine cosine

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Function F(s) F(t) Hyperbolic sine Hyperbolic cosine Exponentially decaying sine wave Exponentially Decaying cosine wave nth root Natural logarithm Bessel Function of the first kind of order n

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Function F(s) F(t) Bessel Function of the second kind of order 0 Modified Bessel Function of the second kind of order n Modified Bessel Function of the second kind of order 0 Heavyside step functin

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*In digital signal and image processing in which the constraints that each output is the effect of only prior inputs In statistics, a weighted moving average is a convolution In GIS, the result of Kernel estimate of intensity function is the convolution of isotropic Gaussian kernel *In physics where there is a linear system with superposition principle *In linear acoustics where an echo is the convolution of the original sound. *In optics, a shadow and light. In probability, the probability of two joint distribution. An out of focus photograph is convolution of sharp image In Artificial reverberation In time-resolved fluoerescence spectroscopy

LEGENDRE POLYNOMIALS:

LEGENDRE POLYNOMIALS 18 September 1752 – 10 January1833) French Mathematician. Work on number theory, abstract algebra and mathematical analysis

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N 0 1 1 x 2 3 4 5

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6 7 8 9 10

Frobenius BESSEL:

Frobenius BESSEL Georg Frobenius 1849 – August 3, 1917 German Mathematician Friedrick Wilhelm Bessel (22 July 1784 – 17 March 1846) German Mathematician

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Plot of the graph of Bessel function of the second kind with various value of

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A simple series Circuit

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Undamped string Mass system

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END GOOD BLESS RAIFU S. AJAYI NUR 2011

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