1. Formation of a Differential Equation – Differential Equations

1

Formation of a Differential Equation

1.1 INTRODUCTION

The general laws of science, engineering, medicine, social sciences, population dynamics and the like where the rate of change of quantities is involved are usually modelled as differential equations. Formation of differential equations, solution and interpretation of the results are of practical interest, especially for engineers and physicists, as they deal with many engineering and physical problems. So, we study in this chapter, methods of solution of ordinary differential equations of the first order and first degree, their applications to Newton's law of cooling the law of natural growth and decay, and orthogonal trajectories.

Further, methods of solution of linear differential equations of second and higher order with constant coefficients are also considered in this chapter.

1.1.1 Differential Equation

An equation expressing a relation between functions, their derivatives and the variables is called a differential equation.

Differential equations are classified into (1) ordinary and (2) partial differential equations.

Ordinary differential equation

A differential equation containing derivatives of a function (or functions) of a single variable is called an ordinary differential equation.

Example 1.1.1

(order, degree)

Partial differential equation

A differential equation containing partial derivatives of a function (or functions) of more than one variable, with each derivative referring to one of the variables is called a partial differential equation.

Example 1.1.2

(order, degree)

With each differential equation, we associate a pair of positive integers (n, m) called the order and degree, respectively, of the differential equation.

Order of a differential equation

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

The general form of an nth order ordinary differential equation in variables x and y is

 

 

where .

Degree of a differential equation

The degree of a differential equation is the degree or power of the highest derivative, when the equation is freed from radicals and fractions in respect of the derivatives, i.e., when the expression F(x, y, y′, y″,..., y(n)) is written as a rational integral algebraic expression in y(n).

If F cannot be expressed in this manner then the degree of the differential equation is not defined.

The order and degree of the differential equations given above are indicated against each of them.

We can also classify differential equations as linear and non-linear. A differential equation is called linear if the sum of the powers of the function and its derivatives in each term is at most equal to unity, and otherwise, non-linear.

Example 1.1.3

  1. The following differential equations are linear.

  2. The following differential equations are non-linear:

1.2 DIFFERENTIAL EQUATIONS

Differential equations arise in many engineering and physical problems. The approach of an engineer to the study of differential equations has to be practical, and so, it consists of

  1. formation of a differential equation from the physical conditions, called modelling;
  2. solution of a differential equation under the initial/boundary conditions; and
  3. the physical interpretation of the results.

1.2.1 Formation of a Differential Equation

An ordinary differential equation is obtained when we eliminate arbitrary constants (also called parameters) from a given relation involving the variables; the order of the equation being equal to the number of the constants to be eliminated.

A partial differential equation is obtained when we eliminate arbitrary constants or functions from a given relation involving the variables/functions.

We consider here only ordinary differential equations and hence by a differential equation we mean an ordinary differential equation.

Differential equations occur in physical, biological, chemical, engineering and geometrical problems, and also in problems relating to economics, ecology, population studies, medicine, etc.

 

 

be a relation involving variables x and y and n arbitrary constants c1,c2,...,cn.

If we differentiate Eq. (1.1) n times with respect to ‘x’ and eliminate the n arbitrary constants between the (n + 1) relations, we get an ordinary differential equation of the form

 

 

where

 

Example 1.2.1   Eliminate a and b from y = ax2 + bx and form the differential equation.

Solution

 

 

Differentiating twice with respect to ‘x’ we have

 

 

 

Example 1.2.2   Form the differential equation by eliminating the constants A and B from

 

 

Solution   Differentiating Eq. (1.6) twice with respect to ‘x

 

 

 

From Eqs. (1.6) and (1.8) we obtain

 

y″ – 4y = 0

 

which is the required differential equation.

Example 1.2.3   Obtain the differential equation by eliminating λ from

 

 

Solution   Differentiating Eq. (1.9) with respect to ‘x’ once

 

 

 

Substituting in Eq. (1.9)

 

 

or   (x + yy′)(xy′ - y) = (a2 - b2)y

 

which is the required differential equation.

Example 1.2.4   Determine the differential equation by eliminating the constants A and B from

 

 

Solution   Differentiating Eq. (1.12) twice with respect to ‘x

 

 

 

From Eqs. (1.12) and (1.14), we obtain

 

 

 

Substituting the values of A and B in Eq. (1.13)

 

 

which is the required differential equation.

Example 1.2.5   Find the differential equation of the family of curves

 

y = Ae2x + Be−2x

 

for different values of A and B

Solution

 

 

Differentiating (1.18) w.r.t. x

 

 

Differentiating (1.19) w.r.t. x

 

 

Eliminating A and B from (1.18) and (1.20) we obtain the differential equation

 

Example 1.2.6   Find the differential equation by eliminating a, b, c from

 

y = ae2x + be-3x + cex

 

Solution

 

 

Differentiating (1.22) w.r.t. x

 

 

Differentiating (1.23) w.r.t. x

 

 

Differentiating (1.24) w.r.t. x

 

 

Now

 

 

∴ The required differential equation is

 

Example 1.2.7   By eliminating the constants a and b from

 

xy = aex + be-x + x2

 

obtain the differential equation

 

Solution

 

 

Differentiating (1.27) w.r.t. x

 

 

Differentiating (1.28) w.r.t. x

 

 

Eliminating a and b from (1.27) and (1.29)

 

Example 1.2.8   Show that Ax2 + By2 = 1 is a solution of

 

 

Solution

 

 

Differentiating (1.31) w.r.t. x

 

 

Differentiating (1.32) w.r.t. x

 

 

Eliminating A and B from (1.32) and (1.33)

 

 

Example 1.2.9   Find the differential equation of the family of curves y = c(xc)2 where c is an arbitrary constant.

 

Solution

 

 

Differentiating (1.34) w.r.t. x

 

 

Squaring (1.35) and dividing it by (1.34)

 

 

Substituting the value of c in (1.35)

 

 

which is the required differential equation.

Example 1.2.10   Find the differential equation of the family of ellipses in the standard form.

Solution   The equation of the family of ellipses in the standard form is

 

 

where a and b are parameters.

Differentiating (1.38) w.r.t. x

 

 

Differentiating (1.39) w.r.t. x

 

 

which is the required differential equation.

Example 1.2.11   Obtain the differential equation of the family of circles with centres on the x-axis and passing through the origin.

Solution   The equation of the family of circles passing through the origin and with centres on the x-axis is

 

 

where ‘a’ is a parameter

Differentiating (1.41) w.r.t. x

 

 

Putting the value of a in (1.41)

 

 

which is the required differential equation.

Example 1.2.12   Find the differential equation of the family of parabolas having vertex at the origin and focii on the y-axis.

 

Solution   The equation of the family of parabolas with vertex at the origin and focii on the y-axis is

 

 

where a is a parameter

Differentiating (1.44) w.r.t. x

 

 

Eliminating a from (1.44) and (1.45)

 

 

which is the required differential equation.

1.2.2 Solution of a Differential Equation

A relation ϕ(x, y) = 0 defining a function y = f(x) in some interval I, which has derivatives f′,f″,...,f(n) such that F(x, f, f′, f″,..., f(n)) = 0, i.e., satisfying the differential equation, is called a solution of the differential equation:

 

F(x, y, y′, y″,..., f(n)) = 0.

General (or complete) solution

A relation ϕ(x, y, c1, c2,...,cn) = 0 containing n independent arbitrary constants ci which is a solution of the differential equation

 

F(x, y, y′, y″,..., y(n)) = 0

 

is called the general (complete) solution of the differential equation.

Particular solution

Any solution obtained from the general solution of a differential equation, by giving particular values to the arbitrary constants in it, is called a particular solution of the differential equation.

Singular solution

A solution ϕ(x, y) = 0 of the differential equation

 

F(x, y, y′, y″,..., y(n)) = 0

 

which is neither a general solution nor a particular solution of it, is called a singular solution of the differential equation. Only some equations have singular solutions.

In the context of differential equations, solution and ‘integral’ have the same meaning; and the general solution is sometimes called the primitive.

Example 1.2.13   y = Ae3x + Be-3x, where A and B are arbitrary constants, is the general solution while

 

y = -2e3x   and   y = 10e-3x

 

are particular solutions of the second order and first degree differentialequation

 

 

It has no singular solution.

Example 1.2.14

y = (x + a)2 is the general solution;

y = x2 is a particular solution; and

y = 0 is a singular solution of the first order and second degree differential equation

 

 

Note that the singular solution y = 0 cannot be obtained from the general solution for any value of the constant ‘a’.

Formation of a differential equation is straight-forward while the solution of a differential equation is not. So, we have to classify them and find methods of solution.

EXERCISE 1.1

Form the differential equation in each of the following cases by eliminating the parameters mentioned against each.

  1. y = ax + bx2                              (a, b)

    Ans: 2y + x2y″ = 2xy

  2. x = A cos(pt + B)                       (A, B)

    Ans: x″ + p2x = 0 (′ denotes differentiation with respect to ‘t’)

  3. Ans:

  4. y = ax2 + bx + c                       (a, b, c)

    Ans: y′″ = 0

  5. (x - h)2 + (y - k)2 = r2                (h, k)

    Ans:

  6. y = ex (A cos x + B sin x)           (A, B)

    Ans: y2 - 2y1 + 2y = 0

  7. Find the differential equation for the family of circles with their centres on the x-axis.

    (Hint: x2 + y2 + 2gx + c = 0 g, c parameters)

    Ans:

  8. Form the differential equation for the family of circles, touching the x-axis at (0, 0).

    (Hint: x2 + y2 - 2fy = 0, f parameter)

    Ans: (y2 - x2)y′ + 2xy = 0

  9. Form the differential equation of all parabolas each having its latus-return = 4a and its axis parallel to the x-axis.

    (Hint: (y - k)2 = 4a(x - h); h, k parameters)

    Ans:

  10. Find the differential equation by eliminating c from y = cx + x3.

    Ans:

  11. Show that y = ax + b ex, where a and b are arbitrary constants, is a solution of .
  12. Show that the differential equation obtained from y = Aeax cos bx + Beax sin bx where A and B are arbitrary constants is y″ - 2ay′ + (a2 + b2)y = 0.
  13. Find the differential equation of the family of circles in the xy plane.

    (Hint: The equation of the circles is x2 + y2 + 2gx + 2fy + c = 0)

    Ans:

  14. Find the differential equation of the family of curves where λ is a parameter.

    Ans: (x2 - a2)y′ = xy

  15. Find the differential equation of all the ellipses with their centres at the origin.

    Ans: xyy′ + x (y′)2 - yy′ = 0

  16. Show that the differential equation of all parabolas having their axes of symmetry coincident with the x-axis is