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11. A unique solution exists in some neighborhood of $x=1$.

12. A unique solution exists in some neighborhood of $x=1$.

13. A unique solution exists in some neighborhood of $x=0$.

14. Existence but not uniqueness is guaranteed in some neighborhood of $x=0$.

15. Neither existence nor uniqueness is guaranteed in any neighborhood of $x=2$.

16. A unique solution exists in some neighborhood of $x=2$.

17. A unique solution exists in some neighborhood of $x=0$.

18. Neither existence nor uniqueness is guaranteed.

19. A unique solution exists in some neighborhood of $x=0$.

20. A unique solution exists in some neighborhood of $x=0$.

21. Your figure should suggest that $y(-4)\approx 3$; an exact solution of the differential equation gives $y(-4)=3+e^{-4}\approx 3.0183$.

22. $y(-4)\approx -3$

23. Your figure should suggest that $y(2)\approx 1$; the actual value is closer to 1.004.

24. $y(2)\approx 1.5$

25. Your figure should suggest that the limiting velocity is about 20 ft/sec (quite survivable) and that the time required to reach 19 ft/sec is a little less than 2 seconds. An exact solution gives $v(t)=19$ when $t=\frac{5}{8}\ln 20\approx 1.8723$.

26. A figure suggests that there are 40 deer after about 60 months; a more accurate value is $t\approx 61.61$. The limiting population is 75 deer.

27. The initial value problem $y\text{′}=2\sqrt{y}\text{,}y(0)=b$ has no solution if $b<0$; a unique solution if $b>0$; infinitely many solutions if $b=0$.

28. The initial value problem $xy\text{′}=y\text{,}y(a)=b$ has a unique solution if $a\ne 0$; infinitely many solutions if $a=b=0$; no solution if $a=0$ but $b\ne 0$.

29. The initial value problem $y\text{′}=3y^{\mathrm{2/3}}\text{,}y(a)=b$ always has infinitely many solutions defined for all x. However, if $b\ne 0$ then it has a unique solution near $x=a$.

30. The initial value problem $y\text{′}=-\sqrt{1-y^2}\text{,}y(a)=b$ has a unique solution if $|b|<1$; no solution if $|b|>1$, and infinitely many solutions (defined for all x) if $b=\pm 1$.

31. The initial value problem $y\text{′}=\sqrt{1-y^2}\text{,}y(a)=b$ has a unique solution if $|b|<1$; no solution if $|b|>1$, and infinitely many solutions (defined for all x) if $b=\pm 1$.

32. The initial value problem $y\text{′}=4x\sqrt{y}\text{,}y(a)=b$ has infinitely many solutions (defined for all x) if $b\geqq 0;$ no solutions if $b<0$. However, if $b>0$ then it has a unique solution near $x=a$.

33. The initial value problem $x^{2}y\text{′}+y^2=0\text{,}y(a)=b$ has a unique solution with initial point (a, b) if $a\ne 0$, no solution if $a=0$ but $b\ne 0$, infinitely many solutions if $a=b=0$.

34. (a) If $y(-1)=-1.2$ then $y(1)\approx -0.48$. If $y(-1)=-0.8$ then $y(1)\approx 2.48$. (b) If $y(-3)=-3.01$ then $y(3)\approx -1.0343$.

If $y(-3)=-2.99$ then $y(3)\approx 7.0343$.

35. (a) If $y(-3)=-0.2$ then $y(2)\approx 2.019$. If $y(-3)=+0.2$ then $y(2)\approx 2.022$. In either case, $y(2)\approx 2.02$. (b) If $y(-3)\approx -0.5$ then $y(2)\approx 2.017$. If $y(-3)\approx +0.5$ then $y(2)\approx 2.024$. In either case, $y(2)\approx 2.02$.