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 c