F03 - Further Differential Equations - Assignment (Solutions)

National Junior College Mathematics Department 2016 2016 – 2017 / H2 FMaths / Further Differential Equations (Assignment Solutions) Page 1 of 5 National Junior College 2016 – 2017 H2 Further Mathematics Further Differential Equations (Assignment Solutions) 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] Solution Mark Scheme 1  13 2d d y xy x                      2 2 2 2 2 2 2 2 d 3 2d 3 d 2 d 3 22 2 6 4 2 0 2 3 2 13 0 2 3 , where 13 2 0 y y xx y y x x y x y x C x y y x C x y C x y C C C                                B1: DE for normal M1: Integrating both sides A1: Correct solution B1: Circle B1: Centre labelled www.KiasuExamPaper.com 145

National Junior College Mathematics Department 2016 2016 – 2017 / H2 FMaths / Further Differential Equations (Assignment Solutions) Page 2 of 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] Solution Mark Scheme 2(i) N represents the carrying capacity of the worm’s population. B1 2(ii) d (1 )d P P kP Pt N    , where  is a positive constant B1 2(iii) (1 ) 0 ( ) 0 0 or PkP P N kPP k N kP P N k            Since the population increases to an equilibrium value in the long run, 0k Nk   M1: Solving for equilibrium values A1: Phase line diagram B1: Converging to the correct equilibrium value 2(iv) When k  , the worm will become extinct in the long run. B1 www.KiasuExamPaper.com 146

National Junior College Mathematics Department 2016 2016 – 2017 / H2 FMaths / Further Differential Equations (Assignment Solutions) Page 3 of 5 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 pow