Ordinary Differential Equation, Partial Differential Equation, Lap Lace Transforms And Vector Analysis. Section – A UNIT – I

Ordinary Differential Equation, Partial Differential Equation, Lap Lace Transforms And Vector Analysis.

Section – A UNIT – I

1. x p² ₂+3xyp+2y² =0

2. Find the particular integrands of

(

^D2+16

)

^{y=cos 4x}

3. Define : Linear differential equation.

4. Eliminate the arbitrary function. z f ^xy z

 

=   

5. Find the particular integrals of ^o+

(

^D2+16

)

^y=e−3x

6. Define Clairaut’s equation.

7. Find the PDE by eliminating the ordinary constant form z px by ^a

= + + b. 8. Solve xp+yq=x.

Section – B

1. Solve. x−yp=ap²

2. Solve.

(

^{D3−D2− +D 1}

)

^{y= +1 x2}

3. Solve.

(

^D3−8D+9

)

^{y=8sin 5x}

4. Solve.

(

^D2+3D2−4

)

^{y=ex+cos 3x}

5. Solve. ^y=xp+x

(

^1+p2

)

^{1/ 2}

6. Solve x=P² +y

7. Solve tan ¹( ) ₂

1

x P P

P

= − +

+ 8. (D₂ −6D+5) y=e^2x

9. Form the P.D.E by eliminating arbitrary constant of equt

2 2

z=ax+by+a b 10. Solve ^{P+ +}

(

^{1 q}

)

^=qz.

11. Solve P² +q² = +x y

Section – C

1. Solve.

(

^D2−2D+4

)

^y=excosx

2. Solve.

(

^D3−2D+4

)

^y=excosx

3. Solve.

(

^D2+1

)

^{y=x e2 2x+xcos}

4. Solve P² +2yp Cotx= y².

5. Solve the equation

(

^{D2 +5D+4}

)

^y=x2+7x+9.

UNIT –II Section – A

1. Eliminate the arbitrary function from ^{z= f x}

(

^{2 +y2}

)

and find the partial differential equation.

2. Solve. q= y q^{2 2}

3. Solve.

(

^d3−3d2+4

)

^y=0

4. Write down claimant’s form.

5. Solve. z= px qy+ +pq

6. Define Lagrange’s linear equation.

7. P(1+q)=qz

Section – B

1. Solve.

(

^{3z−4y p}

)

⁺

(

^{4x−2z q}

)

^=2y−3x

2. Solve. ^{q p}

(

^−sinx

)

^=cosy

3. Solve. ^p

(

^1−q2

)

^{=q z}

(

⁻¹

)

4. Solve. z= px+qy+ 1+p²+q² 5. Solve. pxy+pq+qy= y²

6. p + q =1

7. Solve ^{z= px+qy+2}

( )

^{pq .}

Section – C

1. Solve. (i) z= p² +q² (ii) z= px qy+ −2 pq

2. Determine the surface which satisfies the differential equation

(

^x2−^{a p2}

) (

+ ^{xy az}− ^tan

)

q=xz ay− cot and passes through the curve x²+y² =a², z=0

3. Obtain the complete solution of xp²−ypq+y q³ −y z² =0 4. Solve z= px+qy+ 1+p²+q² .

UNIT – III Section – A

1. Find 1 e^t

L t

 − 

 

 

2. Find

( )

1 1

L s s a

−  

 

 + 

 

3. Find ^{L t}

(

^{2+ +2t 3}

)

4. State final value theorem in Laplace transform.

5. Find ^L

(

^coshat

)

Section – B

1. Find

( )

1 2 2

2

4 5

L s

s s

−  + 

 + + 

 

2. Find L¹ _{2 2}^s ₂ s a b

−  

 + 

 

3. Find

sin2t

L t

 

 

 

4. Evaluate ³

0

tcos te tdt





−

5. Find L te ^−tsint

Section – C

1. Using Leplace transform solve

( ) ( )

2

2 2 3 sin , y 0 0 and y' 0 0 d y dy

y t

dt + dt − = = =

2. Evaluate the following integrads

3

0

( )

t bt

e e

a dt

t

 − −



²

0

( ) sin

t t

b e dt

t

 −



3. Using Leplace transform solve ^{y'' 2 '− y + = +y}

(

^{t 1 2, y 0}

) ( )

⁼⁴

( )

and y' 0 = −2

UNIT-IV Section – A

1. Prove that div =3

2. If f and g are irrotational, show that f g is solenoidal.

3. Find

( )

1

2

1

2 16

L s

−  

 

 + + 

 

4. Find

( )

1

2 2

L s s

−  

 

 + 

 

5. Find   if =xy²+yz³

Section – B

1. Find the directional derivative of  =xy+yz+zxat (1, 2, 0) in the direction of the vector i+2j+2k

2. Prove that ^{div u v}

( )

 =v cur u u cur v^{. /} − ^{. /} 3. Find

( ) ( )

1

2

1

1 2 2

L s s s

−  

 

+ + +

 

 

4. Find

( )

( )

1 2 2

2 1

2 2

L s

s s

−  + 

 + + 

 

5. Prove that ^{div r}

( )

^{+a =o where a} is a constant vector.

Section – C

1. Prove that

( )

i div grad r =n n+1ⁿ

( )

rⁿ⁻²

( )

^{ii curl grad r =0n}

2. Solve the differential equation

2

2 10 24 24

d y dy

y x

dx − dx+ = given that dy 0

y= dx =

when x = 0

3. Prove that ^{div r r}

( )

^{n =}

(

ⁿ⁺³

)

^rn. Deduce that r rⁿ is solenoidal if and only if n = -3

UNIT – V

Section – A

1. State stoke’s Theorem.

2. Define line Integral.

3. Find the unit normal to the surface x³+xyz+z³ =1 at (1,-1,1) 4. If =x z² +e^{y x/} and  =2z y² −xy² find ^{ +}

(

^{ }

)

5. Evaluate

 

₀¹ ₀¹

(

^x2+y2

)

^dxdy

Section – B

1. ^{f =}

(

^{3x2+6y i}

)

^{−14yzj+20xz k2 ,} Evaluate along the path from (0, 0, 0) to (1, 1, 1) .

c f d r



along the path x=t, y=t , z=t^{2 3} 2. Evaluate .

c f n



^{where F =18zi−12j+3yk}and S is the surface of the plane. 2x+3y+6z=12 in the first octant.

3. Find div f and Curl f for ^{f =}

(

^{y2−z2+3yz−2x i}

)

⁺

(

^{3xz+2xy j}

)

(

^{3xy 2xz 2z k}

)

+ − +

4. If  =log ,r where r= +xi yi+zk, Prove that ²

( )

2

logr 1

 = r

5. (i) Define surface integral.

(ii) If f =x i² −xy j and C is the straight line joining the points (0, 0) and (1, 1) find .

c f d r



Section-C

1. Verify Gauss divergence theorem for F =4xi−2y j² +z k² taken over the region bounded by x²+y² =4, z=0 and z=3.

2. Prove that

( )

^a

( )

^{r rn =}

(

ⁿ⁺³

)

^rn

( )

^{b Curl r r}

( )

^{n =0}

3. Verify stake’s theorem for f = y zi² +z x y² +x yk² where S is the open surface of the cube forward by the planes x= a, y=a and z=a,in which the plane Z=-a is cut.