Applied Mathematics -II (Cc)

8/13/2019 Applied Mathematics -II (Cc)

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APPLIED MATHEMATICSII

Course Contents:

Differential Equation, Basic concepts and ideas.Separable equations, Equations reducibleto separable form, Exact differential equations, Integrated factors, Linear first order differential

equations, Bernoullis differential equation. Families of curves, Orthogonal trajectories and

applications of differential equations of first order relevant to engineering systems.

Homogeneous linear differential equations of second order, homogeneous equations with

constant coefficients, the general solutions, initial and boundary value problems, D operator,

complementary functions and particular integrals. Real, Complex and repeated roots of

characteristics equations. Cauchy equation, non homogeneous linear equations. Applications of

higher order linear differential equations. Ordinary and regular points and corresponding series

solutions, Legendres equations and Legendres polynomial. Bessel equations and Bessel

functions of first kind.

Recommended Books:

Advanced Engineering Mathematics, 5th Edition by C.R. Wylie McGraw HillEducation.

Advanced Engineering Mathematics, 8thEdition by Erwin Kreyszig John Wiley & Sons.


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Differential Equation:

DEF: An equation containing the derivatives of one or more dependent variables, with respect

to one or more independent variables, is said to be a differential equation(DE).

For Example:

dy + y cosx=sinxdx

222

0d y dy

+x y =dx dx

2 3 222

1 dy d y

+ =dx dx

are differential equations.

Classification By Type:

If an equation contains only ordinary derivatives of one or more dependent variables with

respect to a single independent variable, it is said to be an ordinary differential equation

(ODE).

For Example;

dy + y cosx=sinxdx

0222

d y dy+x y =

dx dx

+ =dx dy 2x + ydt dt

, (A DE can contain more than one dependent variable)

are ordinary differential equations.

Partial Differential Equation:

An equation involving partial derivatives of one or more dependent variables of two or more

independent variables is called a partial differential equation (PDE).

For Example;

2 22 2

+ = 0u u

x y


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2 22 2

2u u u

x t t

u vy x

are partial differential equations.

Classification By Order:

The order of a differential equation (either ODE or PDE) is the order of the highest

derivative in the equation.

For Example;

32

2+5 4 = x

d y dyy e

dx dx

is a second order ordinary differential equation.

Generalized Form of nth

order ODE:

We can express an nth

orderordinary differential equation in one dependent variable by

the general form

, , , . . . , 0nF x y y y (1)

Where, F is a real valued function of n + 2 variables , , , . . . ,

nx y y y .

Degree Of a DE:

The degree of a differential equation is the degree of the highest order derivative that

appears in the equation.

Classification By Linearity:

An ordinary differential equation , , , . . . , 0nF x y y y is said to be linear in , , , . . . ,

nx y y y . This means that nthorder ODE is linear when (1) is

1

1 1 01 . . .

n n

n nn n

d y d y dya x a x a x a x y g x

dx dx dx

(2)


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Two important specialcases of (2) are

For (n = 1); 1 0dy

a x a x y g xdx

For (n = 2); 2

2 1 02d y dya x a x a x y g xdx dx

Notice that a differential equation which is not linear is called a nonlineardifferential

equation. Nonlinear functions of the dependent variable or its derivatives, such as sinyor ex

cannot appear in a linear equation. Therefore,

1 2 xy y y e ,2

2 sin 0

d yy

dx ,

42

4 0

d yy

dx

are examples of nonlinear first, second and fourth order ordinary differential equations,

respectively. One of the goals in this course is to solve, or find solutions of, differential

equations.

Solution Of an ODE:

A solution (or integral) of a differential equation is a relation between the variables, not

containing derivatives, such that this relation and the derivatives obtained from it satisfy the

given differential equation.

Generation Solution:

A solution of a differential equation which contains the number of arbitrary constants

equal to the order of the equation is called generation solution (or integral) of the differential

equation.

Example: The equation

dyy

dx

has the solutionx

y ce , where the constant c is an arbitrary.

Similarly, the equation2

2 0

d yy

dx has the solutions cosy A x , siny B x and

cos siny A x B x , where A and B are arbitrary constants.


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Particular Solution:

A solution obtained from the general solution by giving particular values to the constants

is called a particular solution (or integral).

INITIAL AND BOUNDARY VALUE PROBLEMS

DEF (Initial Value Problems):

Sometimes it is required to find the solution of a differential equation subject to the

supplementary conditions. If the conditions relate to one value of the independent variable such

as 0 0 0 0and y atx =xy x y x , then they are called theinitial value problems (IVPs).

DEF (Boundary Value Problems):

The problem of finding the solution of a differential equation such that all the associated

supplementary conditions relate to two different values of the independent variable is called a

twopoint boundary value problems (BVPs).

EQUATIONS OF THE FIRST ORDER AND FIRST DEGREE

The problem of finding general solution of a given differential equation will now be considered.

The solution of any differential equation may or may not exist. Even if the general solution

exists, it may not be easy to find. We shall only discuss the methods/ techniques of solutions of

special types of differential equations.

SEPARABLE EQUATIONS

DEF: A differential equation of the type

F(x) G (y) dx + f(x) g (y) dy= 0 (1)

is called an equation with variable separableor simply a separable equation. Equation (1) may

be written as

0G

g

f

F dy

y

ydx

x

x

which can be easily integrated.

Q.NO.1 Solve the differential equations:

i. 0sin2 xydx

dy

ii. 0222 2 dyxxdxyxxy


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iii. 222 22 xyxyxyxdx

dy

iv. 01

12

2

x

y

dx

dy

v. 032

2

3

xxyy

dxdy

vi. 21;0654483 22 ydyxxydxyx vii.

2

10;

4

12

2

y

y

xx

dx

dy

viii.412

,0csccos8 22

ydyxdxy

SOL(ii): Here

0222 2 dyxxdxyxxy

Variable separable

02212 dyxxdxyyx

0221 dyxxdxyx dxyxdyxx 212

dxxx

x

y

dy

2

1

2

By integrating

A2

1

2

dxxx

x

y

dy

Consider,

22 12121

xxxxx (Using Partial Fraction)

Then (A) takes the form, we obtain

dx

xxy

dy

22

1

2

1

2


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cxxy ln2lnln2

12ln

cxxy ln2ln2

12ln

2ln2ln

xx

cy

2

2

xx

cy

which is the general solution of the given differential equation.

SOL(viii): Here

0csccos8 22 dyxdxy

Variable Separable

x

dx

y

dy22 csc

8

cos

dxxdyy 22

sin8sec

dxxdyy

2

2cos18sec2 dxxdyy 2cos14sec2

By integrating, we obtain

dxxdyy 2cos14sec2

cx

xy

2

2sin4tan

cxxy 2sin24tan (A)which is the general solution of the DE.

At12

x ,

4

y ; (A) becomes


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c6

sin212

44

tan

c 13

1

3

c

Hence, the particular solution of the DE is

32sin24tan

xxy

xxy 2sin24

3tan 1

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Applied Mathematics -II (Cc)