1. 1. $y(0.25)\approx 1.55762$; $y(0.25)=1.55760$. $y(0.5)\approx 1.21309$; $y(0.5)=1.21306$. Solution: $y=2e^{-x}$

In Problems 2 through 10 we give the approximation to $y(0.5)$, its true value, and the solution.

1. 2. $1.35867,1.35914;y=\frac{1}{2}{e}^{2x}$

2. 3. $2.29740,2.29744;y=2e^x-1$

3. 4. $0.71309,0.71306;y=2e^{-x}+x-1$

4. 5. $0.85130,0.85128;y=-e^x+x+2$

5. 6. $1.55759,1.55760;u=2\exp (-x^2)$

6. 7. $2.64745,2.64749;y=3\exp (-x^3)$

7. 8. $0.40547,0.40547;y=\ln (x+1)$

8. 9. $1.28743,1.28743;y=\tan \frac{1}{4}(x+\pi )$

9. 10. $1.33337,1.33333;y={(1-x^2)}^{-1}$

10. 11. Solution: $y(x)=2-e^x$.

    x	$\mathit{h}\mathbf{=}0.2$ y	$\mathit{h}\mathbf{=}0.1$ y	Exact y
    0.0	1.00000	1.00000	1.00000
    0.2	0.77860	0.77860	0.77860
    0.4	0.50818	0.50818	0.50818
    0.6	0.17789	0.17788	0.17788
    0.8	$-0.22552$	$-0.22554$	$-0.22554$
    1.0	$-0.71825$	$-0.71828$	$-0.71828$

In Problems 12 through 16 we give the final value of x, the corresponding Runge-Kutta approximations with $h=0.2$ and with $h=0.1$, the exact value of y, and the solution.

1. 12. $1.0,2.99996,3.00000,3.00000$; $y=1+2/(2-x)$

2. 13. $2.0,4.89900,4.89898,4.89898$; $y=\sqrt{8+x^4}$

3. 14. $2.0,3.25795,3.25882,3.25889$; $y=1/(1-\ln x)$

4. 15. $3.0,3.44445,3.44444,3.44444$; $y=x+4x^{-2}$

5. 16. $3.0,8.84515,8.84509,8.84509$; $y={(x^6-37)}^{1/3}$

In Problems 17 through 24 we give the final value of x and the corresponding values of y with $h=0.2$, 0.1, 0.05, and 0.025.

1. 17. $1.0,0.350258,0.350234,0.350232,0.350232$

2. 18. $2.0,1.679513,1.679461,1.679459,1.679459$

3. 19. $2.0,6.411464,6.411474,6.411474,6.411474$

4. 20. $2.0,-1.259990,-1.259992,-1.259993$, $-1.259993$

5. 21. $2.0,2.872467,2.872468,2.872468,2.872468$

6. 22. $2.0,7.326761,7.328452,7.328971,7.329134$

7. 23. $1.0,1.230735,1.230731,1.230731,1.230731$

8. 24. $1.0,1.000000,1.000000,1.000000,1.000000$

9. 25. With both step sizes $h=0.1$ and $h=0.05$, the approximate velocity after 1 second is 15.962 ft/sec (80% of the limiting velocity of 20 ft/sec); after 2 seconds it is 19.185 ft/sec (96% of the limiting velocity).

10. 26. With both step sizes $h=6$ and $h=3$, the approximate population after 5 years is 49.3915 deer (65% of the limiting population of 75 deer); after 10 years it is 66.1136 deer (88% of the limiting population).

11. 27. With successive step sizes $h=1,0.1,0.01,\dots$ the first four approximations to $y(2)$ we obtain are 1.05722, 1.00447, 1.00445and 1.00445. Thus it seems likely that $y(2)\approx 1.00445$ accurate to 5 decimal places.

12. 28. With successive step sizes $h=1,0.1,0.01,\dots$ the first four approximations to $y(2)$ we obtain are 1.48990, 1.46332, 1.46331, and 1.46331. Thus it seems likely that $y(2)\approx 1.4633$ accurate to 5 decimal places.

13. 29. Time aloft: approximately 9.41 seconds

14. 30. Time aloft: approximately 9.41 seconds