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Drill Exercise Differential equations by abhijit kumar jha

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DRILL EXERCISE - 1

1. Find the order and degree (if defined) of the differential equation

542 3

2

d y dy1

dx dx


dx

dynx

dx

yd2

2

.

3. Find the order and degree (if defined) of the differential equation dx

dy = 1 + x + y..

4. Find the order and degree (if defined) of the differential equation dx

dy + xy = cot x.


3

4

4

dx

yd

+ 4

7

dx

dy

+ 6y = 5 cos 3x.

DRILL EXERCISE - 2

1. Obtain the differential equation of all circles of radius r.2. Obtaining the differential equation associated with the primitive, y = c

1e3x + c

2 e2x + c

3 ex,

where c1, c

2, c

3 are arbitrary constants.

3. Find the differential equation of all hyperbolas with coordinate axes as asymptotes.

4. Find the differential equation of all cardioids )cos1(ar .5. Find the order of the differential equation of all tangent lines to the parabola y = x2.6. Find the order and degree (if defined) of the differential equation of all parabolas whose axis is x-axis.

DRILL EXERCISE - 3

1.

dx

dyx13

dx

xdyy 2

2.

dx

dyya

dx

xdyy 2

3. ydx � xdy = xydx4. (1 - x2) (1 - y)dx = xy(1 + y)dy

5. n x x

xdx

n y x

ydy

(sec tan )

cos

(sec tan )

cos

DRILL EXERCISE - 4

1. 1yx

1yx

dx

dy

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2.x y 1 dy x y 1

x y 2 dx x y 2

3. (2x + y+1)dy + (4x + 2y- 1)dx=0

4.22 a

dx

dy)yx(

DRILL EXERCISE - 5

1. (3xy + y2)dx + (x2 +xy)dy = 0

2.xy4x

y2x6

dx

dy2

22

3. 0dy)y6x4(dx)y5x3(

4. (x2 - y2)dx + 2 xy dy = 0, y(1) = 2

5.dx

dyxy

dx

dyxy 22

DRILL EXERCISE - 6

1. (x + 2y � 3)dy = (2x � y + 1)dx 2. (x + y)dx + (3x + 3y � 4)dy = 0

3.dy

dx

x y

y x

6 2 7

3 2 6 4.dy

dx

x y

x y

2 3

2 3

5. (x - y)dy = (x + y + 1)dx

DRILL EXERCISE - 7

1. xsin2xtanydx

dy 2. ( ) ( )x a

dy

dxy x a 3 5

.

3. 0dx

dy)ex()y1( ytan2 1

4. ( ) ( ) /1 2 12 2 1 2 xdy

dxxy x x

5. sin x dy

dx+ 3y = cos x.




DRILL EXERCISE - 8

1. xyyxdx

dy 33 2. 0dyedx)exy2(y xx

3. xcos.ycosxtandx

dyytan 3 4. x

dy

dxy ny xyex

5. yy x x x y sin cos (sin )2

DRILL EXERCISE - 9

1. Solve the differential equation (2 x cos y + y2 cos x)dx + (2y sin x - x2 sin y)dy = 0.

2. Solve the differential equation [y (1 + x -1) + sin y ]dx + (x + log x + x cos y)dy = 0 .

3. Solve the differential equation (x2 sin3 y - y2 cos x)dx + (x3 cos y sin2 y - 2y sin x)dy = 0 .

4. Solve the differential equation y(xy + 1)dx + x(1 + xy + x2y2)dy = 0.

Drill exercise - 10

1. A radioactive substance decays with time such that at any moment the rate of decay of volume isproportional to the volume at that time. Calculate the half-life of the substance, if s20% of it disappearsin 15 years.

2. A yeast grows at a rate proportional to its present size. If the original amount doubles in two hours, inhow many hours will it triple ?

3. A depositor places Rs. 10,000 in a certificate of deposit which pay 6 percent interest per annum,compounded continuously. How much will be in the account at the end of seven years assuming noadditional deposits or withdrawal ?

4. How long will it take a bank deposit to triple in value if interest is compounded continuously at a

constant rate of 4

15 percent per annum?

DRILL EXERCISE - 11

1. A body of unknown temperature is placed in a refrigerator at a constant temperature of00 F. If after 20 minutes the temperature of the body is 400 F and after 40 minutes thetemperature of the body is 200 F, find the initial temperature of the body.

2. A body at a temperature of 500 F is placed in an oven whose temperature is kept at 150 F.If after 10 minutes the temperature of the body is 750 F, find the time required for the bodyto reach a temperature of the of 1000 F.




3. A cup of tea is prepared in a preheated cup with hot water so that the temperature of both the cupand the brewing tea is initially 190o F. The cup is then left to cool in a room kept at a constant 72oF.Two minutes later, the temperature of the tea is 150oF. Determine (a) the temperature of the tea after5 minutes. (b) the time required for the tea to reach 100oF.

4. A body at a temperature of 0oF is placed in a room whose temperature is kept at 100o F. If after 10minutes the temperature of the body is 25oF, find (a) the time required for the body to reach atemperature of 50oF, and (b) the temperature of the body after 20 minutes.

DRILL EXERCISE - 12

1. A tank initially contains 50 lit. of fresh water. Brine contains 2 pounds per lit. ofsalt, flows into the tank at the rate of 2 lit. per minutes and the mixture kept uniform bystirring runs out at the same rate. How long will it take for the quantity of salt in the tank toincrease from 40 to 80 ponds.

2. A tank initially holds 10 lit. of fresh water. At t = 0, a brine solution containing b2

1 of salt per lit. is

poured into the tank at a rate of 2 lit./min., while the well�stirred mixture leaves the tank at the samerate. Find(a) the amount and(b) the concentration of salt in the tank at any time t.

3. A tank contains 40 lit. of solution containing 2 g of substance per lit. . Salt water containing 3 g of thissubstance per lit. runs in at rate of 4 lit./min. and the well stirred mixture runs outat the same rate .Find the amount of substance in the tank after 15 minutes

4. A tank contain 100 lit. of bring made by dissolving 60 lb of salt in water. Salt water containing 1 lb ofsalt per lit. runs in at the rate of 2 lit./min. and the well-stirred mixture runs out at the same rate of3 lit/ min. Find the amount of substance in the tank after 15 minutes.

DRILL EXERCISE - 13

1. Find the orthogonal trajectories of the family of curves x2 � y2 = c2.

2. Find the orthogonal trajectories of the family of curves y2 = 4cx.

3. Find the orthogonal trajectories of the family of curves 2

Cy

x

4. Find the orthogonal trajectories of the family of curves xCy

DRILL EXERCISE - 14

1. Find the family of curves which intersect the family of circles 04gx2yx 22 at right

angle.




2. Find the curve for which the sum of the length of the tangent and subtangent at any of itspoint is proportional to the product of the coordinates of the point of tangency, theproportionality factor is equal to k.

3. Find the equation of the curve passing through the point (e, � e) and which is such that themiddle point of the segment of its normal at any point of the curve to the x-axis, lies on theline x + 2y = 0.

4. Find the equation of the curve intersecting with the x-axis at the point x = 1 and for whichthe length of the subnormal at any point of the curve is equal to the arithmetic mean of thecoordinates of this point (y - x)2 (x + 2y) = 1.




ANSWER KEY

DRILL EXERCISE - 1

1. order = 2, degree = 3 2. order = 2, degree not defined

3. order = 1, degree = 1 4. order = 1, degree = 1

5. order = 4, degree = 3

DRILL EXERCISE - 2

1.

2

2

22

32

dx

ydr

dx

dy1

2. 0y6

dx

dy11

dx

yd6

dx

yd2

2

3

3

3. xy(1) + y = 0 4. dsinrdr)cos1(

5. order = 1

DRILL EXERCISE - 3

1. (y - 3)(1 - 3x) = cx 2. y = c(a + x)(1 - ay) 3. y = cxe-x

4. nx y c y y x( )11

22

1

22 2 2 5. n x x n y y c2 2(sec tan ) (sec tan )

DRILL EXERCISE - 4

1. y � x + log | x + y | = c 2. cx|2)yx(|log2

1y 2

3. (2x + y + 1)2 = 6x + c 4. ycayx

ayxn

2

a

DRILL EXERCISE - 5

1. c)xy2y(x 22 2. (y + 2x)(2y - 3x) = cx

3. (x + 2y)(x + y)2 = c 4. x2 + y2 = 5x 5. y = cey/x

DRILL EXERCISE - 6

1. c5

7y

5

7y

5

1x

5

1x

22

2. x + 3y + 2 log( 2 - x - y) = c




3. 6(y � 1)2 + 4(2x � 3)(y � 1) � 3(2x � 3)2 = c 4. (x + y - 2) = c(y - x)3

5. arctan2 1

2 1

1

22 2y

xnc x y x y

DRILL EXERCISE - 7

1. )cxsin(xsecy 2 2. y = c x a x a( ) ( ) 3 51

2

3. 1 1tan y 2 tan y2xe e c 4. y = c x x( )1 12 2

5.1

3 22

23FHG

IKJ y

xc

xxtan tan

DRILL EXERCISE - 8

1. y2 (x2 + 1 + 2xce ) = 1 2. y-1 ex = c - x2

3. cx2sin2

1x

2

1xsecysec

4. x ny e x cx ( )1

5. y2 = 2

3 2sin

sinx

c

x

DRILL EXERCISE - 9

1. x2 cos y + y2 sinx = c 2. xy + y log x + x sin y = c

3.x y

y c3 3

2

3

sinsin 4. log y =

1 1

2 2 2xy x yc

5. x y Cx2 2

DRILL EXERCISE - 10

1. 46.6 years 2. 3.17 hr. 3. 15219.62 Rs. 4. 20.93 yrs.

DRILL EXERCISE - 11

1. T = 80e-0.035t ; To= 80oF 2. 9.23t;150e100T 100

t029.0

3. (a) 113.90 F (b) 6.95 min.




4. 100e100T t029.0 (a) 23.9 min. (b) 440 F

DRILL EXERCISE - 12

1. 25 ln 3 min. 2. (a) 0.2tQ 5e 5 (b)0.2tQ 1

( e 1)V 2

3. 111.1 gm 4. 56.3 lb

DRILL EXERCISE - 13

1. xy=k 2. )0k(ky2

1x 22

3. x2 � 2y2 = c 4. 2x2 + y = 2c

DRILL EXERCISE - 14

1. x2 + y2 � cy � 4 = 0 2. yk

n c k x 1

12 2 c h

3. ln [(x + y)2 + x2] = 2 1 x y

1 tanx

4. (x - y)2 (x + 2y) = 1


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