F03 - Further Differential Equations - Tutorial

National Junior College Mathematics Department 2016 2016 – 2017 / H2 FMaths / Further Differential Equations (Tutorial) Page 1 of 8 National Junior College 2016 – 2017 H2 Further Mathematics Topic F3: Further Differential Equations (Tutorial) Basic Mastery Questions 1 Solve the following differential equations, expressing y in terms of t: (a) 2d2sec 1d yt yt   (b)  2 d , 0 4d y t yt y t y  2 Solve the following differential equations, expressing y in terms of x: (a)   42 3 4xy x y x   (b)    1 cos , 0 1x y y x y    3 Find the general solutions of the following differential equations. (a) 2 2 d d2 3 2 0d d y y yx x   (b) 2 2 d d16 8 0d d y y yx x   (c) 2 2 d d 4 21 0d d y y yx x   4 Find the general solutions of the following differential equations. (a) 2 2 d d 2 2 10sind d y y y tx x   (b) 2 2 d d 3 2 6e 4d d xy y y xx x    5 Sketch the phase lines for the following differential equations, and state the equilibrium points. (a) 2d 10 3d y y yt    (b) 2 2d (4 )d y y yt   www.KiasuExamPaper.com 234

National Junior College Mathematics Department 2016 2016 – 2017 / H2 FMaths / Further Differential Equations (Tutorial) Page 2 of 8 Practice Questions 6 Experiments indicate that rate at which glucose is absorbed by the body is  times the amount of glucose present in the bloodstream, G. Glucose is injected into a patient’s bloodstream at a constant rate of r units per unit time. Write a differential equation that models the amount of glucose present in the patient’s bloodstream at time t. Solve for G in terms of t, given that the amount of glucose initially present in the patient’s bloodstream is 0G units. Find  lim G t t  . 7 The variable z satisfies the differential equation  d cot π 4 0.d z z x xx    Using the substitution sin yz x , show that  d 4 πd y x yx   . Hence find z in terms of x, given that 2z when π 2x . 8 A first order differential equation of the form    d p q , 0 and 1d ny x y x y n nx    is called a Bernoulli equation. Show that the substitution 1 nu y  reduces the Bernoulli equation into the linear equation        d 1 p 1 qd u n x u n xx     . Hence, solve the differential equation 2d d y xy xyx  . 9 Show that the substitution lnv y reduces the differential equation     d P Q lnd y x y x y yx   to the linear equation      d P Q vd v x x xx  . www.KiasuExamPaper.com 235

National Junior College Mathematics Department 2016 2016 – 2017 / H2 FMaths / Further Differential Equations (Tutorial) Page 3 of 8 Hence solve the equation 2d 4 2 ln 0d yx x y y yx    . 10 Using the graph of the function  f y given below, sketch the phase line for the autonomous equation d f ( )d y yx  . Hence sketch, on a single diagram, the solution curves for (i) (0) 1.2 ,y c  (ii) (0) 0.4 ,y c  (iii) (0) 0.6y c   . 11 When certain kind of chemicals are combined, the rate at which the new compound is form is modelled by the differential equation ( )( )dx k x xdt     , where x is the number of grams of the new compound formed and k > 0 is the constant of proportionality and 0    . (a) Sketch a phase line diagram. Predict the behaviour of x as t  , given that x   when t = 0. (b) Consider the case when   . Use a phase line diagram to predict the behaviour of x as t  when (0)x  . (c) Show that an explicit solution to the above DE in the case when k = 1 and   = 2 is 1( ) 2x t t c   , where c is an arbitrary constant. Find the particular solution satisfying (0) 1.x  Graph this solution curve. Does the behaviour of the solutions as t  agree with your answer to part (b)? 12 The population of a new breed of prawns in a farm is being studied. In an initial survey, the number, P (in thousands), of the prawns at time t months is modelled by the differential equation  d 4d P P P nt    , where n (in thousands) is a positive real constant. Suppose the farmer intends to harvest 1250 prawns per month. Find the least initial number of prawns that he needs to breed in order for him to ensure a high long -term sustainability of the prawn population. c y f (y) O www.KiasuExamPaper.com 236

National Junior College Mathematics Department 2016 2016 – 2017 / H2 FMaths / Further Differential Equations (Tutorial) Page 4 of 8 Solve the differential equation in this case, expressing P in terms of t. 13 For a certain species in a community, there is evidence that there is a minimum population m such that the species will become extinct if the population size of the species falls below m. The condition can be incorporated into the logistic equation by introducing the factor 1 m P     . Thus the population of the species at time t months after an initial survey is modelled by the modified logistic equation d 1 1d P P mkPt b P          , where k and b are positive constants and b not equal m. (i) Give a reason why m b and hence explain why the species will become extinct if the population size of the species falls below m. (ii) Draw a phase line diagram for the differential equation. (iii) Comment on the long -term prospects of the population for various values of the initial population. (iv) Given d d P t has a maximum value at P q , sketch the solution curve for which the initial population is between m and q. 14 Find the general solution of   2 2 d d 1 0d d y y a ayx x    where the real constant a is such that 1a . Find the solution for which y = 1 and d 1d y x  when x = 0. 15 Show that, if y is a function of x and e u x , then 2 2 2 2 2 d d d d d d y y yx x u u  . Given that y satisfies the differential equation 2 2 2 d d 3 5 0d d y yx x yx x   for x > 0. Use the substitution e u x to show that 2 2 d d 2 5 0d d y y yu u   . Hence, find the general solution for y in terms of x. 16 Show, by means of the substitution 4y x z  , that the differential equation     2 2 2 2 2 d d4 8 3 16 12 0d d y yx x x x x yx x      , x > 0, www.KiasuExamPaper.com 237

National Junior College Mathematics Department 2016 2016 – 2017 / H2 FMaths / Further Differential Equations (Tutorial) Page 5 of 8 can be reduced to the form 2 2 d d 0d d z z a bzx x   , where a and b are to be determined. Hence find the general solution for y in terms of x. 17 Obtain the general solution of the differential equation 2 2 d d 4 13 5sin 2 60cos 2 .d d y y y x xx x    Hence, show that if x is large and positive then, whatever the initial conditions,  5sin 2 ,y x   where 1 4tan . 3       18 Show, by means of the substitution cos ,x  that the differential equation     2 2 2 2 2 d d1 2 1 ,d d y yx x n y xx x     where n is a positive integer, may be reduced to 2 2 2 2 d 2sin .d y n y    Hence obtain the general solution of the original solution for 2n in the form       1 1cos cos sin cos f ,A n x B n x x   where A and B are arbitraty constants and  f x is a function of x to be determined. 19 Use the substitution sin y = x to transform the differential equation (*) cos y       2 2 d d t y – sin y 2 d d      t y + 2k cos y      t y d d + sin y = t where k is a real number, into the form  tfcxt xb t xa  d d d d 2 2 where a, b and c are real numbers and f(t) is a function of t to be determined. Hence find the general solution to (*) in the form sin y = g(t) where g(t) is some function of t, in each of the follow