Differential Equation Introduction Booklet - I With the help of this booklet you can easily follow the concepts of differential equations. Dr. Ajay Kumar Tiwari 1/16/2019

Definition

The equation which contains the derivative of y = f(x) i.e. dy/dx, independent variable x and dependent

variable y is known as differential equation.

Formation of differential equation

Differential equation is formed by eliminating arbitrary constants from the given function.

Example 1 Find the differential equation of all straight lines represented by y = mx + c.

Solution:

Differentiating the given equation w.r.t. ‘x’ we get dy

mdx

since in the given equation there are two

arbitrary constants the again differentiating we get 2

20

d y

dx . This is the differential equation of second

order . hence order of differential equation is highest power of derivative.

Concept builder Find the equation of all straight lines having a constant slope ‘m’ and making a positive intercept ‘c’ by

using calculus.

Solution:

Since slope is constant hence 2

20

d y

dx integrating both the w.r.t. ‘x’ we get

dyk

dx given that slope is m

hence k = m , we get dy

mdx

, again integrating we get y = mx + p, ATQ line is passing through (0, c) we

get p = c so the required equation of line is y = mx + c.

Example 2 Form the differential equation of family of lines situated at a constant distance p from the origin.

Solution:

Let the line is y = mx + c, ATQ the length of perpendicular from origin is 21c p m hence the line is

21y mx p m …..(1) differentiating w.r.t. ‘x’ we get dy

mdx

substituting this value in (1) we get

2 22 1

dy dyy x p

dx dx

. This is the required differential equation.

The order of this differential equation is 1 and degree is 2, degree is the power of highest power of

differentiation.

Example 3

Form the differential equation of all circles of constant radius ‘r’ 2 2 2x h y k r .. .(1)

Solution: In this equation there are 2 arbitrary constant h and k. differentiating given equation w.r.t. ‘x’ we get

2 2 0dy

x h y kdx

……(2) again differentiating (1) w.r.t. ‘x’ we get

22

21 0

d y dyy k

dxdx

…..(3)

Eliminating x hand y k from equation (1), (2) and (3) we get

3 3/2

2

2 22 2

2 2

1 1dy dy

dx dxr or r

d y d y

dx dx

This is the required differential equation of order 2 and degree 2.

Solution of differential equation

A solution of differential equation of differential equation is an equation which contains arbitrary

constants as many as the order of differential equation and is called general solution.

Other solutions obtained by giving particular values to the arbitrary constants in general solution are

known as particular solutions.

Methods to find the general solution of differential equation

I. Variables separation

II. Homogenous form

III. Linear form

Example 4

Based on variables separation 2dy

x ydx

Solution:

2

2

1

1

1

1

tan tan( )

tan( )

dy dvput x y v

dx dx

dvv

dx

dvdx c

v

v x c or v x c

x y x c

Example 5

Solve 2 2 2x xy dy x y dx

Solution: Since in this equation sum of powers x and y is 2 and no constant term so this is the homogenous form

For this put dy dv

y vx and v xdx dx

we get 21 1

1 1

dv v dv vv x or

dx v dx v

now applying variables are separable we get

2

2

2 /

12 log 1 log log

1

2 log 1 log log log

log

y x

v dxdv v v x c

v x

cxor v v x c

x y

y cx

x x y

x y cx e

This is the required solution of the given differential equation.

Linear form of differential equation

If the degree of the dependent variable and all derivatives is one, such differential equation is known as

linear differential equation .

1. Differential equation of the form dy

Py Qdx

is known as first order linear equation where P

and Q are functions of x or constant.

2. Differential equation of the form 2

2

d y dyP Qy R

dxdx is known as second order linear

differential equation and P, Q and R are functions of x or constants.

3. Similarly differential equation of order n is defined.

Solution of differential equation of first order is written as

Pdx Pdx

y e Q e dx C

Where Pdx

e is known as Integrating factor. On multiplying both the side by I.F. we can easily find the

solution of given differential equation.

Integrating Factor of differential equation can be determined by different methods which we will

discuss in next notes.

Solution of higher order differential equation is given as

y = Complementary function + Particular Integral

Example 6 Solve y dx – x dy + logx dx = 0

Solution: The given equation is written as

log

2

1log

1. .

log1. . log

1 log .

dx

xx

t

dy yx

dx x x

I F e ex

xSolution of D E y dx put x t

x x

ydxthen dt we get te dt

x x

Integrating by parts and simplifying

y x cx which is required solution

For more problems for practice you can use your class XII book.

Bernoulli’s Equation

Differential equation of the form ndy

py Qydx

is known as Bernoulli’s equation.

For the solution of this equation divide both the sides by yn then

1

1 1 1

1

(1 ) (1 )

n n

dy dvput v

dx n dxy y

dvwe get n vP Q n

dx

Now this is linear form.

Trajectories

The family of curves in a plane is given by f(x, y, c) = 0 depending on single parameter c.

A curve making an fixed angle ϕ at each of its point with a curve of family passing through that point is

known as isogonal trajectory of other family and if θ = π/2 then it is called orthogonal trajectory.

Example 7 Find the orthogonal trajectory of y

2 = 4ax (a being parameter)

Solution:

First of all eliminate parameter from the given equation and its derivative the n replace dy dx

bydx dy

And then integrate you will find required orthogonal trajectory.

2 4 ......(1)

& 2 4 ........(2)

Given y ax

dyy a

dx

Now eliminating ‘a’ from (1) & (2) we get

2

2

2 0

2

dydx dxy x replacing by

dy dx dy

xdx ydy

x y C

This is required orthogonal trajectory of the given trajectory.

Now in Booklet – 2 we will discuss syllabus of Semester – II