DHS Euler's & Improved Euler's Method (9649) Topical Revision

Numerical Methods – Euler Method/Improved Euler Method NUMERICAL METHODS [FM] EULER METHOD / IMPROVED EULER METHOD 1 A solution to the differential equation d 2d y xyx  has y = 1 when x = 1. (i) Use two iterations of Euler method of step size 0.5 to estimate the value of y when x = 2. [3] (ii) Explain whether you would expect this value to be an under-estimate or an over-estimate of the true value. [2] (iii) Explain why it is usually better to improv e accuracy by using the improved Euler method rather than by simply using smaller step sizes in Euler method. [1] [TJC/FM/2018/P1//Q2] 2 The differential equation 2d tan 1,d y yxx  where 1 when 1,yx is to be solved numerically. (i) Carry out two steps of Euler’s method with step length 0. 1 to estimate the value of y when 1.2x , giving your answer to 4 decimal places. [3] (ii) The method in part (i) is now replaced by the improved Euler method. The estimate obtained is 2.0156, given to 4 decimal places. State, wi th a reason, whether this estimate and the one found in part (i) are likely to be overestimates or underestimates of the actual value of y when 1.2x . [2] (iii) Explain why it would be inappropriate to continue the process in part (i) to estimate the value of y when 1.6.x [2] [VJC/FM/2019/P2/Q3] 3 Determine the maximum number of iterations, using the Euler Method, with step size 0.01 on the differential equation 2 d 1,d yxy x  where 0y  when 1x , such that the error between the actual value of y and the approximated value of y is less than 0.001. [6] [NYJC/FM/2018/P1//Q2] H2 Double Math

Numerical Methods – Euler Method/Improved Euler Method 4. The variables x and y are related by the differential equation d f( , )d y x yx  . (i) Taking the initial value as 00()yx y  , explain with the aid of a diagram, how the Euler method can be applied once on the different ial equation to approximate the solution at 0x xh . [3] Given that f, yxy x y x and 00(, )xy  (1.5, 2). (ii) Apply the following methods with a step size 0.5 to estimate y at 2.5x . (a) Euler method [2] (b) Improved Euler method [3] (iii) Comment on the accuracy of the estimates found in part (ii). [2] (iv) State one advantage and one disadvanta ge in using the improved Euler method compared to the Euler method? [2] [RVHS/FM/2018/P1/Q9] 5 A particle moves along a straight line which passes through a fixed point O. It is acted on by two resistive forces, one of which is proportional to its displacement x from O while the other is proportional to its speed v. As a result, the motion of the particle is governed by the differential equation d 72 4 .d vvx vx   Given that 121 when 0,vx estimate the value of v when 1x using (i) one iteration of Euler’s Formula, [2] (ii) one iteration of the improved Euler formula. [2] Hence, explain why v is approximately a linear function of for 0 1.xx  [2] By considering the values of for 0 1,x xv  use the given differential equation to find an expression for this linear function. [2] [VJC/FM/2018/P1//Q5] 6 (i) Show that the substitution 2zy transforms the differential equation  22 0 (1)d12 3 , d yxy x y yx   to 22 d4 6 d1 1 zx z x xx . [2] (ii) Hence find y in terms of x and sketch three members of the family of solution curves. [6]

Numerical Methods – Euler Method/Improved Euler Method (iii) A curve that has a y–intercept at 1 is defined by the differential equation in (1). Use Euler’s method with step length 0.5 to estimate the value of y when 2x , giving your answer to 3 decimal places. [3] (iv) An estimation is considered to be a good estimation if the error is less than 1% of the step length. Determine if the estimation in (iii) is good. [2] [NJC/FM/2019/MYE/P1/Q10] 7 The function  yy x satisfies  3d1 tan .d5 y x x yx  The value of yh is to be found, where h is a small positive number, and  00 .y  (i) Use one step of the improved Euler formula to find an alternative approximation to yh in terms of h. [2] (ii) It can be shown that  yy x satisfies  440.05 0.05 0 tanee d 5 hhx xy hx   . Assume that h is small and hence find another approximation to yh in terms of h. [2] (iii) Discuss the relative merits of these two methods employed to obtain these approximations. [2] [EJC/FM/2018/P1/Q1] 8 A differential equation is given by  d45 2 e 0d xyxx y x    , where 4x . (i) By using the substitution  4ux y , show that the differentia l equation can be reduced to d 2ed xu ux   . Hence, find y in terms of x , given that 0y  when 0x . Hence obtain the value of y when 0.2x . [6] Consider the differential equation   2d45 2 2 0d yxx y x xx      , where 0y  when 0x . (ii) Use the Euler method with step length 0.1 to estimate the value of y when 0.2x . [3] (iii) Use the improved Euler method with step length 0.1 to estimate the value of y when 0.2x . [3] (iv) Comment on your numerical answers for the values of y from parts (i), (ii) and (iii). [2] [HCI/FM/2019/MCT//Q7]

Numerical Methods – Euler Method/Improved Euler Method Answers 1 (i) 4.47 (correct to 3 s.f.) 2 (i) 1.6656 (4d.p) 3 max no of iterations = 20 4 (ii) a. (2.5) 4.96y  b. (2.5) 7.98y  5 (i) 97 (ii) 97.0; 121 24 .vx 6 (ii) 3 2 62 1 x xCy x   (iii) 1.07422 7 (i) tan10 h h (ii)  4 26 0.05 1e 5 2 120 h hhyh   or  26 4111 20 5 2 120 hhyh h      8 (i) ee 4 xx y x    ; 0.0959 (3 s.f.) (ii) 0.0977 3sf (iii)  0.0958 3sf