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

Numerical Methods – Euler’s Method NUMERICAL METHODS [FM] EULER’S METHOD (SOLUTIONS) 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 improve accuracy by using the improved Euler method rather than by simply using smaller step sizes in Euler method. [1] [TJC/FM/2018/P1//Q2] [Solution] (i) d 2d y xyx  Given x1 = 1, y1 = 1 and h = 0.5 When x2 = 1.5,  1 21 d 2( 1 ) ( 1 )2 . 5d 10 . 5 xx yyy h x        When x3 = 2,  2 32 d 2 (1.5)(2.5) 4.47 (correct to 3 s.f.)d 2.5 0.5 xx yyy h x        (ii) xn d d nxx y x  1 3 1.5 3.936 2 2 2(4.46825) = 4.989 Since d d y x is increasing over [1, 2], the value is an under-estimate. Alternatively, 2 2 d1 d 1 dd 22 yy xyxx yx   H2 Double Math

Numerical Methods – Euler’s Method Since x > 0, y > 0 and d 0d y x  , we have 2 2 d 0d y x  Thus the curve is concave upwards (iii) It is usually more efficient to use the improved Euler method as the convergence is faster and so requires fewer computations or lesser computing time.

Numerical Methods – Euler’s Method 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 es timate 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, with a reason, whethe r 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] [Solution] 1 (i) 22 00 d 1t a n , 1 ,1 , f ( , ) 1t a nd y yx xy x y yxx     , By Euler’s Method with h = 0.1: 1 2 10 . 1 f ( 1 , 1 ) 10 . 1 ( 1t a n 1 ) 1.255741 1.25574 0.1f(1.1,1.255741) 1.665560 y y        the value of y when x = 1.2 is 1.6656 (4d.p) . (ii) The improved Euler method usually gives a better estimate than the Euler method. Since 2.0156 > 1.6656, it is likely that the actual value of y is bigger than these estimates. Hence both estimates are likely to underestimate the actual value. (iii) 2f( , ) 1 tanx yy x is discontinuous at 1.572x  .

Numerical Methods – Euler’s Method 3 Determine the maximum number of iter