Maths II Formulas

7/27/2019 Maths II Formulas

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Engineering Mathematic-II

UNIT-I

ORDINARY DIFFERENTIAL EQUATIONS:

2 3 2

( ) ( )

( ) ; ( ) , , , , tan

D y f x

D is aD bD c or aD bD cD d where a b c d are cons ts

=

+ + + + +

1. ODE with constant coefficients: Solution C.F+P.Iy =

Complementary functions: (C.F.)

Sl.No. Nature of Roots C.F

1.1 2m m= ( )

mxAx B e+2.

1 2 3m m m= = ( )2 mxAx Bx c e+ +3.

1 2m m 1 2m x m xAe Be+

4.1 2 3m m m 31 2

m xm x m xAe Be Ce+ +

5.1 2 3,m m m= 3( )

m xmxAx B e Ce+ +

6. m i = ( cos sin )xe A x B x +

7. m i= cos sinA x B x +

Particular Integral: (P.I.)

Type-I

If ( ) 0f x =

then, P.I = 0.

Type-II

If ( )ax

f x e=1

.( )

axP I ea

=


2/13

Replace D by a. If ( ) 0a , then it is P.I. If ( ) 0a = , then diff. denominator

w.r.t D and multiply x in numerator. Again replace D by a. If you get denominator again

zero then do the same procedure.

Type-III

Case: (i) If ( ) sin ( ) cosf x ax or ax=

1. sin (or) cos

( )P I ax ax

D=

Here you have to replace only 2D not D. 2D is replaced by2a . If the

denominator is equal to zero, then apply same procedure as in Type-II.

Case: ii If 2 2 3 3( ) (or) cos (or) sin (or) cosf x Sin x x x x=

Use the following formulas2 1 cos 2

2

xSin x

= ,

2 1 cos 2cos2

xx

+= ,

( )31

sin 3sin sin 34

x x x= , ( )31

cos 3cos cos34

x x x= + and separate 1 2. & .P I P I

Case: iii If ( ) sin cos ( ) cos sin ( ) cos cos ( ) sin sinf x A B or A B or A B or A B=

Use the following formulas:

( )

( )

( )

( )

1( ) in cos ( ) sin( )2

1(ii) cos sin ( ) sin( )

2

1( ) cos cos cos( ) cos( )

2

1( ) sin sin cos( ) cos( )

2

i s A B sin A B A B

A B Sin A B A B

iii A B A B A B

iv A B A B A B

= + +

= +

= + +

= +

Type-IV

If ( ) mf x x=1

.( )

mP I xD

=

1

1 ( )

mxg D

=+

( )1

1 ( ) mg D x

= +


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Here we can use Binomial formula as follows:

i) ( )1 2 31 1 ...x x x x

+ = + +

ii) ( )

1 2 3

1 1 ...x x x x

= + + + +

iii) ( )2 2 31 1 2 3 4 ...x x x x

+ = + +

iv) ( )2 2 31 1 2 3 4 ...x x x x

= + + + +

v)3 2 3(1 ) 1 3 6 10 ...x x x x+ = + +

vi)3 2 3(1 ) 1 3 6 10 ...x x x x

= + + + +

Type-V

If ( )axf x e V= where sin ,cos , mV ax ax x=

1.

( )

axP I e VD

=

Take out axe and replace D by D+a.

1

( )

axe VD a

=+

Type-VI

If ( )nf x x V= where sin ,cosV ax ax=

sin I.P of

cos R.P of

iax

iax

ax e

ax e

=

=

1. ODE with variable co-efficients : (Eulers Method)

The equation is of the form2

2

2( )

d y dyx x y f x

dx dx+ + =

Implies that2 2( 1) ( )x D xD y f x+ + =

To convert the variable coefficients into the constant coefficients

Put logz x= implies zx e=

2 2

3 3

( 1)

( 1)( 2)

xD D

x D D D

x D D D D

=

=

=

whered

Ddx

= andd

Ddz

=


4/13

The above equation implies that ( )( 1) 1 ( )D D D y f x + + = which is O.D.E

with constant coefficients.

2. Legendres Linear differential equation:

The equation if of the form2

2

2( ) ( ) ( )

d y dyax b ax b y f x

dx dx+ + + + =

Put Z = log( )ax b+ , then ( ) zax b e+ =

2 2 2

3 3 3

( )

( ) ( 1)

( ) ( 1)( 2)

ax b D aD

ax b D a D D

ax b D a D D D

+ =

+ =

+ =

whered

Ddx

= andd

Ddz

=

UNIT-II

VECTOR CALCULUS:

1. Vector differential operator is / / /i x j y k z = + +

2. Gradient of / / /i x j y k z = = + +

3. Divergence ofF 31 2

1 2 3,FF F

F where F F i F j F kx y z

= = + + = + +

rr r r rg

4. Curl of F

1 2 3

/ / /

i j k

XF x y z

F F F

= = r

5. If F is a Solenoidal vector then 0F =g

6.

7. If F is an Irrotational vector, then 0XF =

8. Maximum Directional derivative =

9. Directional derivative of in the direction of a aa

= gr

10. Angle between two normals to the surface1 2

1 2

cosn n

n n =

gr r

Where ( )1 1 1

1 1 ( , , )at x y z n = & ( )

2 2 22 2 ( , , )at x y z

n =


5/13

11. Unit Normal vector, n

=

12. Equation of the tangent plane 1 1 1( ) ( ) ( ) 0l x x m y y n z z + + = at 1 1 1( , , )x y z on

the surface ( , , ) 0x y z = . Here l, m, n are coefficients of , ,i j k in .13. Equation of normal line

1 1 1x x y y z z

l m n

= =

14. Work Done =C

F dr g , where dr dxi dyj dzk = + +r

15. If .C

F drr

is independent of the path then curl 0F =

16. In the surface integral,.

dxdydSn k

= r , .dydzdSn i

= r , .dzdxdSn j

= r & dS ndS =

17. Greens Theorem:

If , , ,u v

u vy x

are continuous and one-valued functions in the region R enclosed

by the curve C, thenC R

v uudx vdy dxdy

x y

+ =

.

18. Stokes Theorem:

Let F be the vector point function, defined over an open surface bounded by a

closed curve C, then ( ) xC S

F dr F nds= r

g g

19. Gauss Divergence Theorem:

Let F be a vector point function in a region R bounded by a closed surface S,

then S V

F nds Fdv= g g


6/13

UNIT-III

ANALYTIC FUNCTIONS:

1. Necessary conditions for f(z) to be an analytic function are the

Cauchy Riemann Equations; &u v u v

x y y x

= =

(OR)

X Y Y XU V and U V = = (C-R equations)

2. The Polar form of Cauchy-Riemann Equations:1 1

&u v v u

r r r r

= =

3. The function u(x,y) is said to be harmonic if it satisfies the

Laplace equation :

2 2

2 2 0u u

x y

+ =

4. If the function is harmonic then it should be either real or imaginary part of some

analytic function.

5. Milne Thomson method: for (finding the analytic function f(z) if the real or

imaginary part is given

i) If u is given ( ) ( ,0) ( ,0)x yf z u z dz i u z dz=

ii) If v is given( ) ( ,0) ( , 0)

y x

f z v z dz i v z dz= +

6. To find the analytic function

i) ( ) ; ( )f z u iv if z iu v= + = adding these two

We have ( ) ( ) (1 ) ( )u v i u v i f z + + = +

then ( )F z U iV= + where , & ( ) (1 ) ( )U u v V u v F z i f z = = + = +

Here we can apply Milne Thomson method for F(z).

7. Bilinear transformation is ; 0az b

w ad bccz d

+=

+

8. The cross-ration of 4 pts( ) ( )

( ) ( )1 2 3 4

1 2 3 4 1 2 3 4

2 3 4 1

, , , ( , , , )z z z z

z z z z is z z z zz z z z

=


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9. The cross-ratio is invariant under a bilinear transformation

( ) ( )

( ) ( )

( ) ( )

( ) ( )1 2 3 1 2 3

1 2 3 1 2 3

w w w w z z z z

w w w w z z z z

=

UNIT-IV

COMPLEX INTEGRATION:

1. Cauchys Integral Theorem:

If f(z) is analytic and ( )f z is continuous inside and on a simple closed curve C,

then ( ) 0c

f z dz= .

2. Cauchys Integral Formula:

If f(z) is analytic within and on a simple closed curve C and a is any point inside

C, then( )

2 ( )C

f zdz if a

z a=

3. Cauchys Integral Formula for derivatives:

If a function f(z) is analytic within and on a simple closed curve C and a is any

point lying in it, then( )

2

1 ( )( )

2C

f zf a dz

i z a =

Similarly( )

3

2! ( )( )

2C

f zf a dz

i z a =

, In general

( )

( )

1

! ( )( )

2

n

n

C

n f zf a dz

i z a+=

4. Cauchys Residue theorem:

If f(z) is analytic at all points inside and on a simple closed cuve c, except for a

finite number of isolated singularities 1 2 3, , ,... nz z z z inside c, then

{ }1 1

( ) 2 (sum of the residues of ( )) 2 Re ( ) Re ( )z z z z

C

f z dz i f z i s f z s f z = =

= = + +

.

5. Critical point:


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The point, at which the mapping w = f(z) is not conformal, (i.e) ( ) 0f z = is called

a critical point of the mapping.

6. Fixed points (or) Invariant points:

The fixed points of the transformationaz b

w cz d

+

= + are obtained by putting w = z

in the above transformation, the point z = a is called fixed point.

7. Re { ( )} ( ) ( )z a z a

s f z Lt z a f z=

= (Simple pole)

8. ( )( )1

1

1Re { ( )} ( )

( 1)!

mm

mz a z a

ds f z Lt z a f z

m dz

= =

(Multi Pole (or) Pole of order m)

9. Taylor Series:

A function ( )f z , analytic inside a circle C with centre at a, can be expanded in the

series

2 3( )( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ... ( ) ...

1! 2! 3! !

nnz a z a z a z af z f a f a f a f a f a

n

= + + + + + +

Maclaurins Series:

Taking a = 0, Taylors series reduce to

2 3

( ) (0) (0) (0) (0) ...

1! 2! 3!

z z zf z f f f f = + + + +

10. Laurents Series:0 1

( ) ( )( )

n nn n

n n

bf z a z a

z a

= =

= +

The part1 ( )

n

nn

b

z a

= is called the Principal part

where1

1

1 ( )

2 ( )n n

C

f za dz

i z a +=

&2

1

1 ( )

2 ( )n n

C

f zb dz

i z a =

, the integrals being

taken anticlockwise around 1 2C and C .

11. Isolated Singularity:

A point 0z z= is said to be an isolated singularity of ( )f z if ( )f z is not analytic

at 0z z= and there exists a neighborhood of 0z z= containing no other singularity


9/13

of f(z). Example:1

( )f zz

= . This function is analytic everywhere except at

0z= . 0z = is an isolated singularity of f(z).

12. Removable Singularity:

A singular point 0z z= is called a removable singularity of ( )f z if0

lim ( )f z

z z

exists finitely.

Example:

sinlim ( ) lim 1

0 0

zf z

z

z z

= =

(finite) 0z = is a removable

singularity.

13. Essential Singularity:

If the principal part contains an infinite number of non zero terms, then 0z z= is

known as an essential singularity.

Example:( )

21 1/1/( ) 1 ...

1! 2!z

zzf z e= = + + + has 0z= as an essential

singularity.

CONTOUR INTEGRATION:

14. Type: I

To evaluate the integrals of the form2

0(cos ,sin )f d

Here we shall choose

the contour (closed curve) as the unit circle : 1, ,0 2iC z Put z e = = .

Then2 1

cos2

z

z

+= ,

2 1sin

2

z

iz

= and

1d dz

iz = .

15. Type: II

To evaluate integrals of the form

( )

( )

P x

dxQ x

, Here P(x) and Q(x) are polynomials

in x such that the degree of Q exceeds that of P at least by two and Q(x) does not

vanish for any x. Choose the closed curve C consisting of the following parts.

(i) the semi circle : Z R = in the upper half plane.

(ii) The line segment [-R, R] on the real axis.


10/13

Here ( ) ( ) ( )R

C R

f z dz f x dx f z dz

= + as ( ) 0R f z dz

= .

Where,( )

( )( )

P zf z

Q z=

16. Type: III

The integrals of the form ( )cos (or) ( )sinf x mxdx f x mxdx

where

( )( )

( )

P xf x

Q x= as in Type II.

Use cos . . , sin . .imx imxmx R P of e mx I P of e= = Proceed as in Type II.

UNIT-V

LAPLACE TRANSFORM:

1. Definition:

[ ] 0( ) ( )stL f t e f t dt

=

2.

Sl.No Nature of Roots C.F

1. [ ]1L1

s

2.nL t 1 1

! ( 1)n n

n n

s s+ +

+=

3.at

L e 1

s a

4. atL e 1s a+

5. [ ]sinL at 2 2a

s a+

6. [ ]cosL at 2 2s

s a+


11/13

7. [ ]sinhL at 2 2a

s a

8. [ ]coshL at2 2

s

s a

3. Linear Property: [ ] [ ] [ ]( ) ( ) ( ) ( )L af t bg t aL f t bL g t =

4. First Shifting property:

If [ ]( ) ( )L f t F s= , then

i) [ ]( ) ( )at

s s aL e f t F s = = F(s-a)

ii) [ ]( ) ( )ats s a

L e f t F s +

= = F(s+a)

5. Second Shifting property:

If [ ]( ) ( )L f t F s= ,( ),

( )0,

f t a t ag t

t a

>=


12/13

10. Final value Theorem:

If [ ]( ) ( )L f t F s= , then( ) ( )

s 0

Lt f t Lt sF s

t

=

11.

Sl.No

1.1 1L

s

1

2.1 1

Ls a

ate

3.1 1L

s a

+

ate

4.1

2 2

sL

s a

+

cos at

5.1

2 2

1L

s a

+

1sin at

a

6.1

2 2

sL s a

cosh at

7.1

2 2

1L

s a

1sinh at

a

8.1 1

nL

s

1

( 1)!

nt

n

12. Deriative of inverse Laplace Transform:

[ ] [ ]1 11

( ) ( )L F s L F s

t

=

13. Colvolution of two functions:0

( ) ( ) ( ) ( )t

f t g t f u g t u du = 14. Covolution theorem:

If f(t) & g(t) are functions defined for 0t then [ ] [ ] [ ]( ) ( ) ( ) ( )L f t g t L f t L g t =


13/13

15. Convolution theorem of inverse Laplace Transform:

[ ]1

0

( ) ( ) ( ) ( )

t

L F s G s f u g t u du =

16. Solving ODE for second order differential equations using L.T

i) [ ] [ ]( ) ( ) (0)L y t sL y t y =

ii) [ ] [ ]2( ) ( ) (0) (0)L y t s L y t sy y =

iii) [ ] [ ]3 2( ) ( ) (0) (0) (0)L y t s L y t s y sy y =

17. Laplace Transform:

If f(x+T) = f(x), then f(x) is said to be a of Periodic function with period T.

For such a periodic function, the Laplace Transformation is given by

[ ]0

1( ) ( )

1

Tst

sTL f t e f t dt

e

=

Maths II Formulas

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