CHAPTER 80 NUMERICAL METHODS FOR FIRST-ORDER DIFFERENTIAL EQUATIONS

Transcription

1 CHAPTER 8 NUMERICAL METHODS FOR FIRST-ORDER DIFFERENTIAL EQUATIONS EXERCISE 33 Page 834. Use Euler s method to obtain a numerical solution of the differential equation d d 3, with the initial conditions that when, for the range. to.5 with intervals of.. Draw the graph of the solution in this range. d ' 3 d If initiall and, (and h.), then (') 3 Line in the table below is completed with. and For line, where. and h.: + h('). + (.)(). and (') For line 3, where.: + h('). + (.)(.99).99 and (') For line 4, where.3: + h(').99 + (.)(.599).35 The remaining lines of the table are completed in a similar wa. 3 4, John Bird

2 A graph of the solution of d 3, with initial conditions and is shown below d. Obtain a numerical solution of the differential equation d d +, given the initial conditions that when, in the range (.). Since d d + then d d ( ) or ( ) If initiall and, (and h.), then (') ( ) Line in the table below is completed with, and (') (') For line, where. and h.: + h('). + (.)() and (') ( ).( ). For line 3, where.4: + h('). + (.)(.).9 and (') ( ).4( (.9)) , John Bird

3 For line 4, where.: + h(').9 + (.)(.38).884 and (') ( ).( (.884)).438 For line 5, where.8: + h(') (.)(.438).7934 and (') ( ).8( (.7934)).4984 For line, where.: + h(') (.)(.4984).999 Hence a numerical solution to the differential equation columns in the above table. d d + is given b the first two 3. (a) The differential equation d d + has the initial conditions that at. Produce a numerical solution of the differential equation in the range.(.).5 (b) If the solution of the differential equation b an analtical method is given b 4, determine the percentage error at. (a) d + i.e. d d ' d If initiall. and (and h.), then (') Line in the table below is completed with. and.5. For line, where. and h.: + h(') + (.)(.5).85 and (') For line 3, where.: + h(').85 + (.)(.447) , John Bird

4 and (') For line 4, where.3: + h(') (.)(.35) The remaining lines of the table are completed in a similar wa (b) If 4 then when.,.788 From the Euler method, when.,.7954 Hence, percentage error % % % 4. Use Euler s method to obtain a numerical solution of the differential equation d d, given the initial conditions that when, in the range.(.)3. If the solution of the differential equation is given b, determine the percentage error b 4 using Euler s method when.8 d (a) ' d If initiall. and, (and h.), then (') Line in the table below is completed with. and ()... For line, where. and h.: + h(') + (.)(.). and ('). (.)..99 For line 3, where.4: + h('). + (.)(.99) , John Bird

5 and (').4 (.488) For line 4, where.: + h(') (.)(.555).4849 The remaining lines of the table are completed in a similar wa (b) If, then when.8,.9 4 From the Euler method, when.8,.9878 Hence, percentage error % % % 34 4, John Bird

6 EXERCISE 34 Page 838. Appl the Euler Cauch method to solve the differential equation d d 3 for the range.(.).5, given the initial conditions that when d ' d 3, and h. ( ) 3 3. and from equation (3), page 83, + h( ) P +.(). + C h [ ( ) + f(, P ) ] + h [ ( ) + + (.) [ + P 3 ]. 3 ]. C ( ) Thus the first two lines of the table below have been completed For line 3, h( ) P (.8778).334 P + C h [ ( ) + 3 ] 35 4, John Bird

7 (.) [ ].7. C.7 ( ) The remaining lines of the table are completed in a similar wa. Solving the differential equation in Problem b the integrating factor method gives 3 +. Determine the percentage error, correct to 3 significant figures, when.3 using (a) Euler s method, and (b) the Euler Cauch method. If 3 + then when.3, (a) B Euler s method, when.3,.35 Percentage error %.4% (b) B the Euler Cauch method, when.3, Percentage error %.4% (a) Appl the Euler Cauch method to solve the differential equation d d for the range to.5 in increments of., given the initial conditions that when, (b) The solution of the differential equation in part (a) is given b e. Determine the percentage error, correct to 3 decimal places, when.4 (a) d ' d +, and h. ( ) +. and from equation (3), page 83, + h( ) P +.(). 3 4, John Bird

8 C + h [ ( ) + f(, P ) ] + h [ ( ) + P + ] + (.) [ ] ( ) C Thus the first two lines of the table below have been completed For line 3,. + h( ) P. +.(.).3 C + h [ ( ) + + P ]. + (.) [ ].45 ( ) C The remaining lines of the table are completed in a similar wa (b) If e then when.4, B the Euler Cauch method, when.4,.5884 Hence, the percentage error %.7% , John Bird

9 4. Obtain a numerical solution of the differential equation d d + using the Euler Cauch method in the range (.)., given the initial conditions that when Since d d + then d d ( ) or ( ) If initiall and (and h.), then (') ( ) Line in the table below is completed with, and (') (') For line,., and from equation (3), page 83, + h( ) +.() P C + h [ ( ) + f(, P ) ] + h [ ( ) + ( P ) ] + (.) [ +.( ()) ] ( ) ( C.98 ).( (.98)).9 Hence, line is completed in the above table. For line 3,.4, and from equation (3), page 83, + h( ) P.98 +.(.9) + C h [ ( ) + f(, P ) ] + h [ ( ) + ( P ) ] (.) [.9 +.4( (.94)) ] , John Bird

10 ( ) ( C ).4( (.9547)) Hence, line 3 is completed in the above table For line 4,., and from equation (3), page 83, + h( ) P C + h [ ( ) + f(, P ) ] + h [ ( ) + ( (.34377) P ) ] (.) [ ( ( )) ] ( ) ( C ).( (.84854)) Hence, line 4 is completed in the above table For line 5,.8, and from equation (3), page 83, + h( ) P C + h [ ( ) + f(, P ) ] + h [ ( ) + ( ( ) P ) ] (.) [ ( ( )) ] ( ) ( C ).8( ( )).4944 Hence, line 5 is completed in the above table For line,., and from equation (3), page 83, + h( ) P + C h [ ( ) + f(, P ) ] + h [ ( ) + ( (.4944) P ) ] 39 4, John Bird

11 (.) [ ( ( )) ].8938 Hence, line is completed in the above table. 4 4, John Bird

12 EXERCISE 35 Page 843. Appl the Runge Kutta method to solve the differential equation: d 3 for the range d.(.).5, given the initial conditions that when d d 3., and since h., and the range is from to.5, then.,.,.3,.4 and Let n to determine : f (, ); since d 3, f (, ) 3 d. k f (, ) h h.. 3. k f +, + k f +, + () f (.5,.5) h h.. k f +, + k f +, + (.479) f (.5, ) 5. k f ( h, hk ) f (.,.( ) ) f (., ) n + n h k + k + k k and when n : + { } h k k k k + { } + { } 3 4. (.479) ( ) {.7778 } n n n , John Bird

13 Lines and have now been completed in the above table Let n to determine : f (.,.4545); since d 3, f (.,.4545) d.. k f (, ) h h.. k f +, + k f. +, (.87777) f.5, h h.. k f +, + k f. +, (.797) ( ).797 f (.5,.85 ) k f ( h, hk ) f (..,.4545.(.474) ) f (.,.597) n + n h k + k + k k and when n : + { } h k k k k + { } (.797) (.474) { } { }.7 Line 3 has now been completed in the above table. In a similar manner 3, 4 and 5 can be calculated.. Obtain a numerical solution of the differential equation: d + using the Runge Kutta d method in the range (.)., given the initial conditions that when If d d + then d d and d ( ) d., and since h., and the range is from to., then 4 4, John Bird

14 .,.4,.,.8and Let n to determine : f (, ); since d ( ), f (, ) ( ) d. k f (, ) h h.. 3. k f +, + k f +, + () f (., ) h h.. k f +, + k f +, + (.) 4. 3 f (.,.99 ) 5. k f ( h, hk ) f (.,.(.98) ) 4 3.( )..(.98) f (.,.984). n + n h k + k + k k and when n : + { } h k k k k.( (.984)).9 + { } + { } Let n to determine :. k f (, ) (.) (.98).9 {.588 } f (.,.98395); since d ( ), d f (.,.98395).( (.98395)).958 h h.. k f +, + k f. +, (.958) f.3, ( (.979).7775 h h.. k f +, + k f. +, (.7775) ( ) f (.3, ).3( ( )) k f ( + h, + hk ) f (. +., ( ) ) 4 3. n + n + { } f (.4,.989).4( (.989)) h k + k + k k and when n : h k k k k + { } , John Bird

15 { (.7775) + ( ) } {.998 } This completes the third row of the table below. In a similar manner 3, 4 and 5 can be calculated. 3. (a) The differential equation: d + has the initial conditions that at. d Produce a numerical solution of the differential equation, correct to decimal places, using the Runge Kutta method in the range.(.).5 (b) If the solution of the differential equation b an analtical method is given b: 4, determine the percentage error at. (a) If d + then d + d d., and since h., and the range is from. to.5, then.,.,.3,.4 and Let n to determine : f (, ); since, d + d, f (, ) +.5. k f (, ) 3. h h...95 k f +, + k f. +,. + (.5) f (.5,.95) , John Bird

16 h h.. k f +, + k f. +,. + (.4595) 4. 3 f (.5, ) k f ( h, hk ) f (..,..(.45498) ) f (., ) n + n h k + k + k k and when n : + { } h k k k k + { } {.5 + (.4595) + (.45498) + (.4783) }.. + { } Let n to determine :. k f (, ).8547 f (.,.8547); since d + d, 3. f (.,.8547) h h.. k f +, + k f. +, (.47954) 4. 3 f(.5, ) h h.. k f +, + k f. +, (.34844) f (.5, ) 5. k f ( + h, + hk ) f (., (.35837) ) f (., ) n + n h k + k + k k and when n : + { } h k k k k + { } , John Bird

17 { (.34844) + (.35837) } { }.788 This completes the third row of the table below. In a similar manner 3, 4 and 5 can be calculated and the results are as shown (b) If 4 when.,.788 B the Runge Kutta method, when.,.788 also Hence, there is no error 4 4, John Bird