Drill Exercise Differential equations by abhijit kumar jha DRILL EXERCISE - 1 DRILL EXERCISE - 2. e x, where c 1

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1 DRILL EXERCISE -. Find the order and degree (if defined) of the differential equation d y d y. Find the order and degree (if defined) of the differential equation n. 3. Find the order and degree (if defined) of the differential equation = + + y. Find the order and degree (if defined) of the differential equation + y = cot. 5. Find the order and degree (if defined) of the differential equation 4 d y y = 5 cos 3. DRILL EXERCISE -. Obtain the differential equation of all circles of radius r.. Obtaining the differential equation associated with the primitive, y = c e 3 + c e + c 3 e, where c, c, c 3 are arbitrary constants. 3. Find the differential equation of all hyperbolas with coordinate aes as asymptotes. Find the differential equation of all cardioids r a( cos ). 5. Find the order of the differential equation of all tangent lines to the parabola y =. 6. Find the order and degree (if defined) of the differential equation of all parabolas whose ais is -ais.. y 3. y a y 3. y = y ( - ) ( - y) = y( + y) 5. DRILL EXERCISE - 3 n(sec tan ) n(sec y tan ) cos cosy DRILL EXERCISE - 4. y y

2 . y y y y 3. ( + y+) + (4 + y- )=0 ( y) a DRILL EXERCISE - 5. (3y + y ) + ( +y) = 0. 6 y 4y 3. ( 3 5y) (4 6y) 0 ( - y ) + y = 0, y() = 5. y y DRILL EXERCISE - 6. ( + y 3) = ( y + ). ( + y) + (3 + 3y 4) = y 7 3y 6 y 3 y 3 5. ( - y) = ( + y + ) DRILL EXERCISE - 7. y tan sin. ( a) 5 3y ( a). tan y 3. ( y ) ( e ) 0 ( ) y ( ) / 5. sin + 3y = cos.

3 DRILL EXERCISE y 3 y. y(y e ) e tan y tan cos y.cos yny ye 5. yy sin cos (sin y ) DRILL EXERCISE - 9. Solve the differential equation ( cos y + y cos ) + (y sin - sin y) = 0.. Solve the differential equation [y ( + - ) + sin y ] + ( + log + cos y) = Solve the differential equation ( sin 3 y - y cos ) + ( 3 cos y sin y - y sin ) = 0. Solve the differential equation y(y + ) + ( + y + y ) = 0. Drill eercise - 0. A radioactive substance decays with time such that at any moment the rate of decay of volume is proportional to the volume at that time. Calculate the half-life of the substance, if s0% of it disappears in 5 years.. A yeast grows at a rate proportional to its present size. If the original amount doubles in two hours, in how many hours will it triple? 3. A depositor places Rs. 0,000 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 no additional deposits or withdrawal? How long will it take a bank deposit to triple in value if interest is compounded continuously at a constant rate of 5 percent per annum? 4 DRILL EXERCISE -. A bo of unknown temperature is placed in a refrigerator at a constant temperature of 0 0 F. If after 0 minutes the temperature of the bo is 40 0 F and after 40 minutes the temperature of the bo is 0 0 F, find the initial temperature of the bo.. A bo at a temperature of 50 0 F is placed in an oven whose temperature is kept at 5 0 F. If after 0 minutes the temperature of the bo is 75 0 F, find the time required for the bo to reach a temperature of the of 00 0 F.

4 3. A cup of tea is prepared in a preheated cup with hot water so that the temperature of both the cup and the brewing tea is initially 90 o F. The cup is then left to cool in a room kept at a constant 7 o F. Two minutes later, the temperature of the tea is 50 o F. Determine (a) the temperature of the tea after 5 minutes. (b) the time required for the tea to reach 00 o F. A bo at a temperature of 0 o F is placed in a room whose temperature is kept at 00 o F. If after 0 minutes the temperature of the bo is 5 o F, find (a) the time required for the bo to reach a temperature of 50 o F, and (b) the temperature of the bo after 0 minutes. DRILL EXERCISE -. A tank initially contains 50 lit. of fresh water. Brine contains pounds per lit. of salt, flows into the tank at the rate of lit. per minutes and the miture kept uniform by stirring runs out at the same rate. How long will it take for the quantity of salt in the tank to increase from 40 to 80 ponds.. A tank initially holds 0 lit. of fresh water. At t = 0, a brine solution containing b of salt per lit. is poured into the tank at a rate of lit./min., while the well stirred miture leaves the tank at the same rate. 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 g of substance per lit.. Salt water containing 3 g of this substance per lit. runs in at rate of 4 lit./min. and the well stirred miture runs outat the same rate. Find the amount of substance in the tank after 5 minutes A tank contain 00 lit. of bring made by dissolving 60 lb of salt in water. Salt water containing lb of salt per lit. runs in at the rate of lit./min. and the well-stirred miture runs out at the same rate of 3 lit/ min. Find the amount of substance in the tank after 5 minutes. DRILL EXERCISE - 3. Find the orthogonal trajectories of the family of curves y = c.. Find the orthogonal trajectories of the family of curves y = 4c. C 3. Find the orthogonal trajectories of the family of curves y Find the orthogonal trajectories of the family of curves y C DRILL EXERCISE - 4. Find the family of curves which intersect the family of circles y g 4 0at right angle.

5 . Find the curve for which the sum of the length of the tangent and subtangent at any of its point is proportional to the product of the coordinates of the point of tangency, the proportionality factor is equal to k. 3. Find the equation of the curve passing through the point (e, e) and which is such that the middle point of the segment of its normal at any point of the curve to the -ais, lies on the line + y = 0. Find the equation of the curve intersecting with the -ais at the point = and for which the length of the subnormal at any point of the curve is equal to the arithmetic mean of the coordinates of this point (y - ) ( + y) =.

6 ANSWER KEY DRILL EXERCISE -. order =, degree = 3. order =, degree not defined 3. order =, degree = order =, degree = 5. order = 4, degree = 3 DRILL EXERCISE -. 3 d y r 3 d y d y. 6 6y y () + y = 0 ( cos )dr r sin d 5. order = DRILL EXERCISE - 3. (y - 3)( - 3) = c. y = c(a + )( - ay) 3. y = ce - n( y) c y y 5. n (sec tan ) n (sec y tan y) c DRILL EXERCISE - 4. y + log + y = c. y log ( y) c 3. ( + y + ) = 6 + c a y a n c y y a DRILL EXERCISE - 5. (y y) c. (y + )(y - 3) = c 3. ( + y)( + y) = c + y = 5 5. y = ce y/ DRILL EXERCISE y y c. + 3y + log( - - y) = c

7 3. 6(y ) + 4( 3)(y ) 3( 3) = c ( + y - ) = c(y - ) 3 5. arctan y nc y y DRILL EXERCISE - 7. y sec ( sin c). y = c( a) ( a) tan y tan y e e c F 3 5 y = c( ) I 3 y HG 3 K J tan c tan DRILL EXERCISE - 8. y ( + + ce ) =. y - e = c - 3. sec ysec sin c ny e ( ) c 5. y = c sin 3 sin DRILL EXERCISE - 9. cos y + y sin = c. y + y log + sin y = c 3. sin y y sin c log y = y y c 5. y C DRILL EXERCISE years. 3.7 hr Rs yrs. DRILL EXERCISE -. T = 80e t ; T o = 80 o 0.09t F. 00e 50; t (a) F (b) 6.95 min. T 00

8 0.09t T 00e 00 (a) 3.9 min. (b) 44 0 F DRILL EXERCISE - 0.t. 5 ln 3 min.. (a) Q 5e 5 (b) Q ( e 0.t ) V 3.. gm 56.3 lb DRILL EXERCISE - 3. y=k. y k(k 0) 3. y = c + y = c DRILL EXERCISE y cy 4 = 0. y k n c k c h 3. ln [( + y) + ] = y tan ( - y) ( + y) =