π (sinπ ) + cos π + C = C = 0 C = 1

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1 National Mu Alpha Theta Diff EQ page Mississippi, 00 Differential Equations Topic Test: Complete Solutions Note: For each problem, where there is no choice (e), assume (e) none of the above.. State the order of the differential equation: ( ) sin st b) nd c) rd d) th Ignore the rd power. The equation remains of order.. The solution to the differential equation d d 0 is e + C b) C c) d) cannot be solved d d 0 d d ln C ln ln + C e + ln C e e C cos d + d 0;The graph passes through ( π,0). Solve ( ) cos sin + b) C sin cos + π c) sin + cos + d) no real valued solution eists answer: c cos d ( ) d (integrate b parts on left) sin + cos + C π (sinπ ) + cos π + C C 0 C + d d 0. Solve the differential equation: ( ) C e b) ( ) e C c) Use v v + dv d v+ v v ( ) ( ) Ce d) dv d C v vdv v v ln + v + ln v ln C (v ) e v C e C ( ) e C Solve the DE : ( ) ( ) + + C b) + C c) + ( ) ( ) + d + + d 0 d d) C F + ( + ) d + + C

2 National Mu Alpha Theta Diff EQ page Mississippi, 00 ( + ) d + + C ( + ) d + + g() + + g( ) + g ( ) g( ) + C so sin d cos d 0. Solve the DE ( ) C cos C b) answer: c Let u sin ( + u ) d - d 0 d + ud - du 0 d du ud d d du d - u ln e so integ. fac e mult b - d du u d sin C ln c) sin C d) not solvable d du ( u) u so d d d d ( u ) + u C u C replace u sin C 7. What is the velocit of a projectile at an altitude of 8000 feet after it was fired directl upward from the ground with a muzzle velocit of 000 feet per second? (g ft/sec) b) 770. c) 9. d) see end 8. A certain tpe of glass is such that a slab inch thick absorbs one-quarter of the light which starts to pass through it. How thin must a pane be made to absorb onl % of the light? (all answers are in inches) b) 0.0 c) 0.08 d) 0.0 see end 9. A certain radioactive material loses mass at a rate proportional to the mass present. If the material has a half-life of 0 minutes, what percent of the original mass is epected to remain after 0.9 hours? 8% b) 9% c) 7% d) 98% Answer: b

3 National Mu Alpha Theta Diff EQ page Mississippi, 00 Solution: r rt rt M M0e e e ln(0.) 0.r.8 r so M.8(0.9) e 0.87 or 8.7% M 0 d e 0. If the marginal cost () of producing a certain item () is + +, what is the cost d of producing one item if there is a fied cost of $.00? $.7 b) $7. c) $8. d) $9. e e + + d + + C & C 7 so + + e $7.. Solve 0. see end Ce C e + b) Ce C e + c) Ce Ce + d) not solvable. What is the time required for one dollar to double when invested at the rate of % per annum compounded continuousl? 0.9 rs b).8 rs c).8 rs d) 8.9 rs answer: c rt A Ae since A A.0t e t.8 rs 0 0 d. Solve the differential equation: d. + c b) +c c) + c d) + c E) NOTA see end

4 National Mu Alpha Theta Diff EQ page Mississippi, 00. The temperature inside a house is 70 F. A thermometer is taken from the house and placed outside. The outside air is 0 F. After minutes, the thermometer reads F. What is the thermometer temperature after 7 minutes? 9 F b) F c) 9 F d) 7 F see end. A pipe 0 cm in diameter contains steam at 00 C. It is covered with asbestos cm thick. The thermal conductivit, k, is cal/cm deg sec. The outside surface is at 0 C. Find the heat loss per hour from a meter length of pipe. (answers are in cal/hr) 80 b) 8,00 c) 8,000 d),80,000 at end. What integrating factor would make the differential equation ( ) eact? b) ( ) d + d 0 b 8) d + d 0 M n 0 so int. factor d e e ln c) d) d + d 0 7. Water flows down a river at the rate 9 + t million ft /da, t das after a rain. How much water will flow past a given point during the first das after a rain? (answer in million ft ) 9. b) 8.8 c) d) dw 9 t + W 9 + t 0 W 9t+ t A solution of the differential equation d d is

5 National Mu Alpha Theta Diff EQ page Mississippi, 00 8 b) + 8 c) - d) e) - d d + C + C + C C or 9. If a car accelerates from 0 to 70 mph in 0 sec, what distance does it travel in those 0 sec? (assume acceleration is constant and 0 mph88ft/sec) ft b) ft c) ft d) 07 ft mph 88 ft / s sec 0mph () 0. 0.t + C v t t 0, v 0 C 0 () 0.t s t.t + C t 0, s 0 C 0 ().t s t t 0 s ft 0. The growth size of an animal population at time t is denoted b dp 0.00P (000 P). The population is growing fastest initiall b) at the carring capacit c) when P00 d P d) when > 0 answer: c grows fastest when its derivative 0 and when is concave down

6 National Mu Alpha Theta Diff EQ page Mississippi, 00 P 0.00P d P 0.00P 0 when P is concave down for all P. Functions g and h are twice differentiable such that h() ln(g()) and h // () f()/(g()) Find f(). g()g // ()- g / () b) g()g // () g / () c) g()[g // ()] g / () d) g()g // () [g / ()] see end. Given ds t t. If s 0 when t, then what is the value of s when t 0? d) 7 b) 8 7 c) t t ds ( t t ) s t + C s 0, t 0 7 so 0 + C C for t 0 7 s. Find the general solution for cannot be done b) + + Ce c) Ce d) + Ce see end +. The motion of a particle on the -ais has acceleration when t. Find () t. d t t. It is stationar at

7 National Mu Alpha Theta Diff EQ page 7 Mississippi, 00 b) t + t t t + 8t+ 7 c) t + 8t d) t t + t see end. The general solution of d d is a famil of circles b) parabolas c) hperbolas d) lines passing through the origin d d d d ln ln + C C. If radium decomposes at a rate proportional to the amount present, then the amount R left after t ears, if R 0 is present initiall and k is a negative constant of proportionalit, is given b kt Rkt 0 R R0kt b) R R0 e c) R R0 + kt d) R e dr cr ln R ct + K R Ke ct since K R 0 when t 0, then R R0 e ds 7. Given π b) π when t 0, and s. Find t when s. sin s c) d) π ct π s ds csc π cot s t+ C, π t 0, s C 0 π s cot t t π π 8. In 970, the earth's population was. billion. If the rate of increase is % per ear, the ear in which the population will reach 0 billion will be closest to which of the following ears? 00 b) 0 c) 00 d) 00

8 National Mu Alpha Theta Diff EQ page 8 Mississippi, P separate& integrate t C 0.0t P Ce 0.billion ln(.8) t t 0. e ears 9. Use Euler's method and steps with 0. for the differential equation to find 0 and 0.. an approimation for, when ( ). b).97 c).87 d).07 X Y i 0 0.() (.)() (.)() (.78)() Which of the following differential equations is NOT logistic? P P P d b) 0.0 (00 ) d c) dr d) 0.(0 ) R all others are of the form r( C )

CHAPTER 6 Differential Equations

CHAPTER 6 Differential Equations Section 6. Slope Fields and Euler s Method.............. 55 Section 6. Differential Equations: Growth and Deca........ 557 Section 6. Separation of Variables and the Logistic