Differential Equations Notes and explanation for First year Engineering students
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PRESENTATION ON DIFFERENTIAL EQUATION
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CONTENTS : INTRODUCTION OF DIFFERENTIAL EQUATIONS. + INVENTION OF DIFFERENTIAL EQUATIONS ORDER AND DEGREE OF DIFFERENTIAL EQUATIONS, FORMATION OF DIFFERENTIAL EQUATIONS. ORDINARY DIFFERENTIAL EQUATIONS (ODE). SOLUTION OF DIFFERENTIAL EQUATIONS. METHODS FOR SOLVING ODE • REAL APPLICATIONS OF DIFFERENTIAL EQU s
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What are Differential Equations Calculus, the science of rate of change, was invented by Newton in the investigation of natural phenomena. Many other types of systems can be modelled by writing down an equation for the rate of change of phenomena: bandwidth utilisation in TCP networl
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A mathematical equation that relates a function O with its derivatives is called differential equation, 0 the function usually represent physical quantities, 0 derivatives represent its rate of change differential equation defines a relationship O between the two.
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History of Differential Equations Origin of differential equations Who invented idea Bacl
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Origin of differential equations mathematics history of differential equations traces the development of differential equations form calculus, itself independently invented by English physicist Isaac Newton and German mathematician Gottfried Leibniz. The history of the subject of differential equations in concise form a synopsis of the recent article "The History of Differential Equations 1670-1950".
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Sir Isaac Newton and Gottfried Leibniz
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Slope and Rate of change Change in Y Slope = Change in X We can find an Average slope between two points. But how do we find the slope at a point? There is nothing to measure! But with derivatives we use a small difference ... ...then have it shrink towards zero. hange in Y hangein x o slope : o 4 24 —115 24 average slope 15 Change in Y Slope = Change in X Ay Ax
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Differential calculus real tim e video
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Differential Equation A Differential Equation is an equation with a function and one or more of its derivatives differential equation (derivative) dx dy Example: an equation with the function y and its derivative dx
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derivative differential 3 dx 2 Y dx -1 y This is a differential equation because it has 'derivative' components in it This is a differential equation because it has 'differential' components in it This is NOT a differential equation because it does not have Idifferential' nor 'derivative' components in it This is NOT a differential equation because it is not a form of equation (no 'equal' sign) even though it has 'derivat•vel component i
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Form ation of Differential Equations the family of straight lines represented by Y = mx dx dx 0 m = tano x is a equation of the first order,
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Form ation of Differential Equations Assume the family of curves represented by Y = Acos (x + B) where A and B are arbitrary constants. dy —A Sin (x + B) dx d2y and 2 dx —A cos (x -k B) [Differentiating (i) w.r.t. x] [Differentiating (ii) w.r.t. x]
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Form ation of Differential Equations d2y 2 dx [Using d2y 2 dx is a differential equation of second order Similarly, by eliminating three arbitrary constants, a differential equation of third order is obtained. Generally eliminating n arbitrary constants, a differential equation of nth order is obtained.
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Order and Degree Next we work out the Order and the Degree: Order 2 Degree 3 3 dx2 dx Order The Order is the highest derivative is it a first derivative? a second derivative? etc): Example: — + Y2 5x It has only the first derivative dydx so is "First Order"
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Example: dx2 = sin(x) This has a second derivative — , so is "Order 2" dX2 Example: d3y dy 3 dx dx dy This has a third derivative — which outranks the so is 'Order : dx dX3
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Degree The degree is the exponent of the highest derivative. Example: 2 + y 5x2 The highest derivative is just dy/dx, and it has an exponent of 2, so this is "Second Degree" In fact it isa First Order Second Degree Ordinary Differential Equation Example: d3y dy ) 2 + Y = 5x2 dX3 The highest derivative is d3y/dx3, but it has no exponent (well actually an exponent of 1 which is nct shown), so this is "fiest Degree". (The exponent of 2 on dy/dx does not count, as it is not the highest derivative). So it is a Third Order First Degree Ordinary Differential Equation
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Solving . We solve it when we discover the function y (or set of functions y), There are many "tricl
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Solution of a Differential Equation The solution of a differential equation is the relation between the variables, not taking the differential coefficients, satisfying the given differential equation and containing as many arbitrary constants as its order is, For exam pie: y = Acosx - Bsinx d2y 2 dx
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General Solution If the solution of a differential equation of nth order contains n arbitrary constants, the solution is called the general solution. Y = Acosx - Bsinx is the general solution of the differential equation Y = —Bsinx d2y 2 dx is not the general solutüon as it contains one arbitrary constant.
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Particular Solution A solution obtained by giving particular values to the arbitrary constants in general solution is called particular solution. y = 3cosx-2sinx d2y 2 dx is a particular solution of the differential equation .
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SOLUTION OF DIFFERENTIAL EQUATION,
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Variable Separable The first order differential equation dy f(x,y) Is called separable provided that f(x,y) can be written as the product of a function of x and a function of y
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Suppose we can write the above equation as dy = g(x)h(y) dx We then say we have "separated" the variable, By taking h(y) to the LHS, the equation becomes.
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1 -----—dy = g(x)dx On Integrating, we get the solution as 1 --— dy = f g(x)dx + c Where c is an arbitrary constant,
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Separation of Variables Separation of Variables is a special method to solve some Differential Equations A Differential Equation is an equation with a function and one or more of its derivatives differential equation (derivative) dx dy Example: an equation with the function y and its derivative dx
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When Can I Use it? dx - 5xy Separation of Variables can be used whem All the y terms (including dy) can be moved to one side of the equation, and All the x terms (including dx) to the other side. : 5xdx
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Homogeneous Differential Equations A Differential Equation is an equation with a function and ane or more of its derivatives differential equation (derivative) dy dx 5xy Example: an equation with the function y and its derivative dx Here we look at a special method for solving "Homogeneous Differential Equations"
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Homogeneous Differential Equations A first order Differential Equation is Homogeneous when it can be in this form: dy dx We can solve it using Separation of Variables but first we create a new variable v = v = Y is also y=vx And dy = d (vx) dx dv (by the Product Rule) dx dx dx dx dv Which can be simplified to dx dy dv Using y = vx and we can solve the Differential Equation, =v+x dx
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APPLICATION OF DIFFERENTIAL EQUATIONS
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NEWTON'S LAW OF O COOLING„, states that the rate of change of the temperature of an object is proportional to the difference between its own temperature and th ambient temperature (i.e. the temperature of its surroundi g
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Applications on Newton' Law of Cooling: Investigations. • It can be used to determine the time of death. Computer manufacturing. • Processors. • Cooling systems. solar water heater. calculating the surface area of an object.
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Why Are Differential Equations Useful? o In our world things change, and describing how they change often ends up as a Differential Equation: " Rabbits" Exam ple : The more rabbits we have the more baby rabbits we get. Then those rabbits grow up and have babies too! The population will grow faster and faster.
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The important parts of this are: the population N at any time t, the growth rate r the population's rate of change —N dt Let us imagine some actual values: the population N is 1000 the growth rate r is 0.01 new rabbits per week for every current rabbit The population's rate of change —N is dt then = 10 new rabbits per week
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But that is only true at a specific time, and doesn't include that the population is constantly increasing. Remember: the bigger the population, the more new rabbits we get! bo it is better to say the rate of change (at any instant) is the growth rate times the population at that instant: dt And it is a Differential Equation, because it has a function NCt) and its derivative. And how powerful mathematics is! That short equation says "the rate of change of the population over time equals the growth rate times the population "
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Sim ple harmonic motion In Physics, Sirnple Harrnonic Motion is a type of periodic m otion where the restoring force is directly proportional to the displacem ent.An exam ple of this is given by mass on a spring. Example: Spring and Weight A spring gets a weight attached to it: the weight is pulled down by gravity, >the tension in the spring increases as it stretches, >then the spring bounces back up, >then back down, up and down, again and again. Describe this with mathematics! tt('i 10.0 20.0 10.0 40.0 — 90.0 40.0
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Let us see a video on Newton's law of cooling
Discussion
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