Ch7 Differential Equations BMQ Solutions (RVHS)

2022 JC1 H2 Mathematics 9788 RVHS Mathematics Department Ch 7 Differential Equations (Solutions to Tutorial and Assignment) 1 CHAPTER 7 DIFFERENTIAL EQUATIONS Tutorial Solutions Basic Mastery Questions 1. Solve the equation 2d 1d yxx x  . 2 2 d 1d d1 for 0d 1ln 2 yxx x y xxxx y x x C        2. Solve the equation d 1d y yx  , given that y < 1 and that 0y when 0x . 1 dd1 ln 1 1 e , since 1 0 1 e where e xC xC yxy y x C yy y A A                 When x = 0, y = 0 then A = 1. 1e xy  3. Use the substitution y vx , where v is a function of x , to solve the differential equation d 3d yx x yx  , given that 0y when 2x . Prove that, in the general case, 2 2 d3 d y xx  . d 3 (1)d yx x yx    Let y = vx dd dd yv xvxx   Substitute into (1): 2 d 3d vx xv x vxx When x = 2, y = 0: 0 3ln 2 C 3ln 2C   3 ln 3 ln 2 3 ln 2 y x x x xx   From (1), 2 2 d d d 3d d d y y yx x x x   2 2 d3 d y xx

2022 JC1 H2 Mathematics 9788 RVHS Mathematics Department Ch 7 Differential Equations (Solutions to Tutorial and Assignment) 2 d3 d 3dd 3ln 3ln v xx vx x v x C y xcx        4. When a body moves freely under gravity, we know that 2 2 d d s gt  , where s is the height of the body above the ground at time t and g is the acceleration due to gravity. Assuming that 210 g m s  , find the general solution for the second-order differential equation. Given that the initial velocity is 15 ms  and initial height is 3 m, find the particular solution for the differential equation. 2 2 10ds dt  (Velocity) 10ds tcdt   ---(1) 25s t ct d    ---(2) (Ans.) When t = 0, 5ds dt  : 5c (from (1)) When t = 0, s = 3 : 3d (from (2)) Particular solution: 25 5 3s t t  

2022 JC1 H2 Mathematics 9788 RVHS Mathematics Department Ch 7 Differential Equations (Solutions to Tutorial and Assignment) 3