The format for the first eight answers is this: $(x(t),y(t))$ at $t=0.2$ by the Euler method, by the improved Euler method, by the Runge-Kutta method, and finally the actual values.

1. 1. $(0.8800,2.5000)\text{,}(0.9600,2.6000)\text{,}(1.0027,2.6401)$, $(1.0034,2.6408)$

2. 2. $(0.8100,-0.8100)\text{,}(0.8200,-0.8200)\text{,}(0.8187,-0.8187)\text{,}(0.8187,-0.8187)$

3. 3. $(2.8100,2.3100)\text{,}(3.2200,2.6200)$, $(3.6481,2.9407)\text{,}(3.6775,2.9628)$

4. 4. $(3.3100,-1.6200)\text{,}(3.8200,-2.0400)\text{,}(4.2274,-2.4060)\text{,}(4.2427,-2.4205)$

5. 5. $(-0.5200,2.9200)\text{,}(-0.8400,2.4400)$, $(-0.5712,2.4485)\text{,}(-0.5793,2.4488)$

6. 6. $(-1.7600,4.6800)\text{,}(-1.9200,4.5600)$, $(-1.9029,4.4995)\text{,}(-1.9025,4.4999)$

7. 7. $(3.1200,1.6800)\text{,}(3.2400,1.7600)$, $(3.2816,1.7899)\text{,}(3.2820,1.7902)$

8. 8. $(2.1600,-0.6300)\text{,}(2.5200,-0.4600)\text{,}(2.5320,-0.3867)\text{,}(2.5270,-0.3889)$

9. 9. At $t=1$ we obtain $(x,y)=(3.99261,6.21770)$($h=0.1$) and $(3.99234,6.21768)$ ($h=0.05$); the actual value is $(3.99232,6.21768)$.

10. 10. At $t=1$ we obtain $(x,y)=(1.31498,1.02537)$($h=0.1$) and $(1.31501,1.02538)$ ($h=0.05$);the actual value is $(1.31501,1.02538)$.

11. 11. At $t=1$ we obtain $(x,y)=(-0.05832,0.56664)$($h=0.1$) and $(-0.05832,0.56665)$ ($h=0.05$);the actual value is $(-0.05832,0.56665)$.

12. 12. We solved $x\text{′}=y\text{,}y\text{′}=-x+\sin t\text{,}x(0)=y(0)=0$. With $h=0.1$ and also with $h=0.05$ we obtain theactual value $x(1.0)\approx 0.15058$.

13. 13. Runge-Kutta, $h=0.1$: about 1050 ft in about 7.7 s

14. 14. Runge-Kutta, $h=0.1$: about 1044 ft in about 7.8 s

15. 15. Runge-Kutta, $h=1.0$: about 83.83 mi in about 168 s

16. 16. At $40°$: 5.0 s, 352.9 ft; at $45°$: 5.4 s, 347.2 ft; at $50°$: 5.8 s, 334.2 ft (all values approximate)

17. 17. At $39.0°$ the range is about 352.7 ft. At $39.5°$ it is 352.8; at $40°$, 352.9; at $40.5°$, 352.6; at $41.0°$, 352.1.

18. 18. Just under $57.5°$

19. 19. Approximately 253 ft/s

20. 20. Maximum height: about 1005 ft, attained in about 5.6 s; range: about 1880 ft; time aloft: about 11.6 s

21. 21. Runge-Kutta with $h=0.1$ yields these results: (a) 21400 ft, 46 s, 518 ft/s; (b) 8970 ft, 17.5 s;(c) 368 ft/s (at $t\approx 23$).