AIEEE Concepts®: Differential Equations

Differential Equation

A differential equation is an equation which involves derivatives of a dependent variable with respect to an independent variable, i.e. it is a relation between x, y and derivatives of y w

.r .t x. Some examples of differential equations are:

(i)

(ii)

(iii)

(iv)

(v)

(vi)

.

Order and Degree of a Differential Equation

The order of the highest differential coefficient appearing in the differential equation is said to be the order of the differential equation. The highest power of the highest order

differential coefficient appearing in the polynomial form of the differential equation is called the degree of the differential equation.

Solution of Differential Equations

A function y = f (x) is a solution of a differential equation, if the substitution of f (x) and its derivative(s) in the differential equation reduces it to an identity. A solution of a differential equation is also called an integral of the differential equation. In order to obtain the solution of a differential equation, we integrate it as many times as the order of the differential equation.

If we have a differential equation of order 'n' then by solving a differential equation we mean to get a family of curves with n parameters whose differential equation is the given differential equation.

Note:
Not all differential equations that we come across have solutions.)

First order differential equations with separable variables.

The differential equation be of the form

= f (x, y) where f(x, y) denotes a function in x and y. Above equation is separable if it can be expressed in the form

where

M(x), N(y) are real valued functions of x and y respectively.Then, we have

N(y) dy = M(x) dx. Integrating both sides we get the solution viz.

.

Note:
The constant of integration is introduced immediately after the integration is carried out. Otherwise, a part of the solution may be lost.

Differential Equations Reducible to the Separable Variable Type

Sometimes differential equation of the first order cannot be solved directly by variable separation. By some substitution we can reduce it to a differential equation with separable variables.

First Order Differential Equations of Homogeneous Nature

The differential equation of the form

is said to be homogeneous differential equation. These equations are solved by putting y = vx, where v

v (x) is a function of x, so that

and the differential equation becomes

. After integrating, we replace v by

and obtain the general solution.

Equations Reducible to the Homogenous Form

Equations of the form

(aB

Ab) can be reduced to a homogenous form by changing the variables x, y to X, Y by writing x = X + h, y = Y + k; where h, k are constants to be chosen so as to make the given equation homogenous. We have

.
After integrating, we replace t by Ax + By and obtain the general solution.

First Order Linear Differential Equation
The differential equation of the form

where P, Q are functions of x alone, is called a linear differential equation. Multiplying by

on both sides, we get

, where

'(x) =

or y =

which is the required solution of the given differential equation.

The factor

is called the integrating factor.

Note:
The importance of introducing arbitrary constant c, immediately after integration. Otherwise, the term

would be lost.

Extended Form of Linear Differential Equations

Sometimes a differential equation is not linear but it can be converted into a linear differential equation by some suitable substitution.

Bernoulli's equation

. It is a linear differential equation.

General Form of Linear Differential Equation:

A general form of the linear differential equation is f '(y)

+ P(x) f(y) = Q(x).

Here, we write f(y) = t . The differential equation becomes

.

General Form of Variable Separation
If we can write the differential equation as

f₁(f₂(x)) d(f₂(x)) + f₃ (f₄(x)) d(f₄(x)) + ... + f_{2n - 1} (f_2n(x)) d (f_2n(x)) = 0,

where f₁, f₂, ... ,f_2n are real valued functions, then each term can be easily integrated separately. For this the following derivatives must be remembered:

(i) d(x + y) = dx + dy (ii) d(xy) = y dx + x dy

(iii) d

(iv) d

(v) d(ln xy) =

(vi)

(vii)

(viii)

(ix)

A special type of second-order differential equation

The differential equation y =

= F₁ (x) + Ax + Bis called second order differential equation.

Orthogonal Trajectory

Any curve which cuts every member of a given family of curves at right angle is called an orthogonal trajectory of the family.

Procedure for finding the orthogonal trajectory

(i) Let f(x, y, c) = 0 be the equation of the given family of curves, where c is an arbitrary parameter.

(ii) Differentiate (1), w.r.t.x and then eliminate c i.e form a differential equation.

(iii) Substitute -

for

in the above differential equation. This will give the differential equation of orthogonal trajectories.

(iv) By solving this differential equation we get the required orthogonal trajectories.

Application of Differential Equations

In solving Rate of Change Problems

Geometrical Applications:

We also use differential equations for finding family of curves for which some conditions involving the derivatives are given. Equation of tangent at a point (x, y) on the curve y = f(x) is

given by Y - y =