F03 - Further Differential Equations - Assignment

National Junior College Mathematics Department 2016 2016 – 2017 / H2 FMaths / Further Differential Equations (Assignment) Page 1 of 2 National Junior College 2016 – 2017 H2 Further Mathematics Topic F3: Further Differential Equations (Assignment) Name: _______________________________ Class: 1fm2__ Subject Tutor: ______________ 1 The normal at any point on a certain curve always passes through the point  2,3 . Form a differential equation to express this property and hence find the equation of the family of curves that possess this property. Sketch a typical member of this family. [5] 2 Spruce budworm is a serious pest in eastern Canada and northern Minnesota. It consumes the leaves of coniferous tree and excess consumption can damage and kill the tree. In the absence of predators, the worm’s population, P (in millions), satisfies the logistic growth, d 1d P P kPt N      . One scientist proposed that the worm’s predators eat the wor m at a rate proportion al to the worm’s population. (i) Explain the significance of the constant N in this model. [1] (ii) Write down the differential equation of the population growth of the worm in the presence of predators. [1] (iii) Given that the population of worm increases towards an equilibrium value in the long run, draw the phase line diagram and comment on the stability of the equilibrium values. [3] (iv) Under what condition will the population of the worms decrease and become extinct eventually. [1] [2014 H3 Prelim/HCI/Modified] 3 The current in a particular electrical circuit is described by the equation 2 2 d d 25 100 170sin 20 ,d d I I I tt t   where I is the current in amperes and t is the time in seconds after the power source is switched on. Find the solution for which d 0d I It   when 0.t  [8] Given that 1I I as t , find the maximum value of 1.I [2] Suggested Duration: 55min Time Spent www.KiasuExamPaper.com 143

National Junior College Mathematics Department 2016 2016 – 2017 / H2 FMaths / Further Differential Equations (Assignment) Page 2 of 2 4 A 2 kg object stretches a vertical spring 80cm beyond its natural length to reach the equilibrium position. Now the spring is stretched further by 5cm and released from th at position with zero velocity. (a) Assume that the air resistance is negligible. Write down the differential equation that models the behaviour of the spring. Define all the variables used in the equation. [2] (b) It is known th at air resistance is proportional to the velocity of the object with the proportionality constant being k . (i) Take 1k  . Find the exact position of the spring at any time t. [4] (ii) Describe the damping effect of the air resistance. [2] (c) In order for the system to be critically damped, it is now placed in a viscous liquid with damping coefficient  . What should be the value of  ? You may assume the buoyancy is negligible. [2] www.KiasuExamPap