ORDINARY DIFFERENTIAL EQUATIONS

Introduction

Motivation and Early Developments –

Just we look at some real life situations where

·        The motion of projectile, rocket, satellite and planets

 

·        the charge or current in the electric circuit

 

·        the conduction of heat on a rod or in a slab

 

·        the vibrations of a wire or membrane etc

 

Differential equations have applications in many branches of physics, physical chemistry etc. In this chapter we study some basic concepts and learn how to solve simple differential equations.

 

 We give below some relations between the rate of change and unknown functions that occur in real life situations.

 (a) The rate of change of y with respect to x is directly proportional to y:

                                                                                                                                     

 

 

(b) The rate of change of y with respect to x is directly proportional to the product of y2 and x:

         

 

(c) The rate of change of y with respect to x is inversely proportional to y:

 

 

(d) The rate of change of y with respect to x is directly proportional to y2 and inversely proportional to :

 

·         A differential equation is an equation in which some derivatives of the unknown function occur. In many cases the independent variable is taken to be time.

     The subject of differential equations was invented along with calculus by Newton and Leibniz in order to solve problems in geometry and physics. It played a crucial part in the development of Newtonian physics by the Bernoulli family, Euler, and others.

     Differential Equation

                          A differential equation is any equation which contains at least one derivative of an unknown function, either ordinary derivative or partial derivative.

                   Examples:                    + 5x = cosx                                                                    

                                                          x

 

     Order of a differential equation

                             The order of a differential equation is the highest order derivative present in the differential equation.

                   Examples:                    has order 1.

    Degree of a differential equation

                          If a differential equation is expressible in a polynomial form, then the integral power of the highest order derivative appears is called the degree of the differential equation.

                 Examples:     The degree of             is not defined since the differential equation cannot be expressed as a polynomial equation in the derivatives.

·        If a differential equation is not expressible to polynomial equation form having the highest order derivative as the leading term then that the degree of the differential equation is not defined.

 

Classification of Differential Equations

Ordinary Differential Equation                      

                       If a differential equation contains only ordinary derivatives of one or more functions with respect to a single independent variable, it is said to be an Ordinary Differential Equation (ODE).

                                        ,          are some examples of ordinary differential equations.

 

Partial Differential Equation

                An equation involving only partial derivatives of one or more functions of two or more independent variables is called a Partial Differential Equation (PDE).

                          and   are some examples of partial differential equations.

 

·         A general linear ordinary differential equation of order n is any differential equation that can be written in the following form

                                                      an(x)y(n) + an-1(x)y(n-1) +………………….+ a1(x) + a0y = g(x)

where the coefficients an(x) , a0(x) , a1(x) and g (x) are any function of independent variable x (including the zero function)

Note –

(1) The important thing to note about linear differential equations is that there are no products of the function, y (x), and its derivatives and neither the function nor its derivatives occur to any power other than the first power.

(2) No transcendental functions – (trigonometric or logarithmic etc) of y or any of its derivatives occur in differential equation.

(3) Also note that neither the function nor its derivatives are “inside” another function, for instance,  or ey′.

 (4) The coefficients a0(x), a1(x),………… , an-1(x) and g(x) can be zero or non-zero functions, or constant or non-constant functions, linear or non-linear functions. Only the function, y(x), and its derivatives are used in determining whether a differential equation is linear.

·        A nonlinear ordinary differential equation is simply one that is not linear.

·        Non linear functions of the dependent variable or its derivatives, such as sin y or ey′ cannot appear in a linear equation.

 

Homogeneous Equation and Non - Homogeneous Equation

                    If g (x) = 0 in an(x)y(n) + an-1(x)y(n-1) +………………….+ a1(x) + a0y = g(x) , then the above equation is said to be homogeneous, otherwise it is called non-homogeneous.

Formation of Differential Equations

      Formation of Differential equations from Physical Situations

              Model 1: (Newton’s Law)

             According to Newton’s second law of motion, the instantaneous acceleration a of an object with constant mass m is related to the force F acting on the object by the equation F = ma. In the case of a free fall, an object is released from a height h(t) above the ground level , Then, the Newton’s second law is described by the differential equation

 

Model 2: (Population Growth Model)

                  The population will increase whenever the offspring increase. For instance, let us take rabbits as our population. More number of rabbits yield more number of baby rabbits. As time increases the population of rabbits increases. If the rate of growth of biomass N (t) of the population at time t is proportional to the biomass of the population, then the differential equation governing the population is given by

 = rN

Model 3: (Logistic Growth Model)

               The rate at which a disease is spread (i.e., the rate of increase of the number N of people infected) in a fixed population L is proportional to the product of the number of people infected and the number of people not yet infected:

 

Formation of Differential Equations from Geometrical Problems

 Differentiate the given equation successively n times, getting n differential equations. Then eliminate n arbitrary constants from (n +1) equations made up of the given equation and n newly obtained equations arising from n successive differentiations. The result of elimination gives the required differential equation which must contain a derivative of the nth order.

·        The result of eliminating one arbitrary constant yields a first order differential equation and that of eliminating two arbitrary constants leads to a second order differential equation and so on.

 

Solution of Ordinary Differential Equations

Solution of DE

                                A solution of a differential equation is an expression for the dependent variable in terms of the independent variable(s) which satisfies the differential equation.

·         There is no guarantee that a differential equation has a solution.

 

General solution

             The solution which contains as many arbitrary constants as the order of the differential equation is called the general solution.

Remark

              The general solution includes all possible solutions and typically includes arbitrary constants (in the case of an ODE) or arbitrary functions (in the case of a PDE).

 

Particular solution

               If we give particular values to the arbitrary constants in the general solution of differential equation, the resulting solution is called a Particular Solution.

Example - Show that x2 + y2 = r2 where r is a constant, is a solution of the differential equation .

      The given equation contains exactly one arbitrary constant. So, we have to differentiate the given equation once.

Differentiate (1) with respect to x, we get

           
 

  which implies

 

Hence

x2 + y2 = r2  is a solution of the differential equation .

Solution of First Order and First Degree Differential Equations

         Variables Separable Method

                            Finding a solution to a first order differential equation will be simple if the variables in the equation can be separated. An equation of the form

f1(x)g1(y)dx + f2(y)g2(x)dy = 0 is called an equation with variable separable or simply a separable equation.

 

Remarks

      1. No need to add arbitrary constants on both sides as the two arbitrary constants are combined together as a single arbitrary constant.

      2. A solution with this arbitrary constant is the general solution of the differential equation

·        “Solving a differential equation” is also referred to as “integrating a differential equation”, since the process of finding the solution to a differential equation involves integration.

 

Example: (1 + x2)  = 1 + y2

          Given that    (1 + x2)  = 1 + y2

The given equation is written in the variables separable form

                                                                                              =

Integrating both sides of (2), we get tan-1y = tan-1x + c

                            But tan-1y - tan-1x = tan-1  

Substitution Method

 Let the differential equation be of the form

(i) If a ≠ 0 and b≠ 0 then the substitution ax+ by +c =z reduces the given equation to the variables separable form.

 (ii) If a = 0 or b = 0, then the differential equation is already in separable form.

Homogeneous Form or Homogeneous Differential Equation

Homogeneous Function of degree n

A function f (x, y) is said to be a homogeneous function of degree n in the variables x and y if, f (tx , ty) =tnf(x , y) for some  for all suitably restricted x, y and t . This is known as Euler’s homogeneity.

f (x , y) is always expressed in the form g  or g .

Homogeneous Differential Equation

An ordinary differential equation is said to be in homogeneous form, if the differential equation is written as

Theorem -

If M (x, y ) dx + N (x,  y) dy = 0 is a homogeneous equation, then the change of variable y =v x, transforms into a separable equation in the variables v and x .

 

First Order Linear Differential Equations

A first order differential equation of the form

where P and Q are functions of x only.

 y

Here  is known as the integrating factor (I.F).

Example – Solve

Given
  

This is a linear differential equation. Here P = 2; Q = .

  

Thus, I.F.         
 

Hence the solution is

 

 

Applications of First Order Ordinary Differential Equations

              The subject of differential equations has vast applications in solving real world problems. The solutions of the differential equations are used to predict the behaviors of the system at a future time, or at an unknown location.

 

      Population growth

                  Let x (t) be the size of the population at any time t . Although x (t) is integer-valued, we approximate x (t) as a differentiable function and techniques of differential equation can be applied to determine x (t). Assume that population grows at a rate directly proportional to the amount of population present at that time. Then, we obtain

·        The population increases exponentially with time. This law of population growth is called Malthusian law.

Radioactive decay

           The nucleus of an atom consists of combinations of protons and neutrons. Many of these combinations of protons and neutrons are unstable, that is the atoms decay or transmute into the atoms of another substance. Such nuclei are said to be radioactive.

 

·        A single differential equation can serve as a mathematical model for many different phenomena.

Newton’s Law of cooling/warming

         According to Newton’s law of cooling or warming, the rate at which the temperature of a body changes is proportional to the difference between the temperature of the body and the temperature of the surrounding medium the so-called ambient temperature.

Mixture problems

                A substance S is allowed to flow into a certain mixture in a container at a constant rate, and the mixture is kept uniform by stirring. Further, in one such situation, this uniform mixture simultaneously flows out of the container at another rate. Now we seek to determine the quantity of the substance S present in the mixture at time t.

Problem on Applications

 

A tank contains 1000 litres of water in which 100 grams of salt is dissolved. Brine (Brine is a high-concentration solution of salt (usually sodium chloride) in water) runs in a rate of 10 litres per minute, and each litre contains 5grams of dissolved salt. The mixture of the tank is kept uniform by stirring. Brine runs out at 10 litres per minute. Find the amount of salt at any time t .

Sol  :      

Let x (t) denote the amount of salt in the tank at time t . Its rate of change is

Now, 5 grams times 10litres gives an inflow of 50grams of salt. Also, the out flow of brine is 10 litres per minute. This is 10 / 1000 = 0.01of the total brine content in the tank. Hence, the outflow of salt is 0.01 times x (t), that is 0.01x (t).

 Thus the differential equation for the model is

This can be written as

Integrating on Both sides,

log |x – 5000| = -0.01t + logC

 

 

Initially, when t = 0, x = 100, so 100 = 5000 + C .Thus, C = −4900 .


Hence, the amount of the salt in the tank at time t is  x = 5000 – 4900e-0.01t