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]