1. $y_0=3,y_1=3+3x,y_2=3+3x+\frac{3}{2}{x}^2,y_3=3+3x+\frac{3}{2}{x}^2+\frac{1}{2}{x}^3,y_4=3+3x+\frac{3}{2}{x}^2+\frac{1}{2}{x}^3+\frac{1}{8}{x}^4$; $y(x)=3e^x$

3. $y_0=1,y_1=1-x^2,y_2=1-x^2+\frac{1}{2}{x}^4,y_3=1-x^2+\frac{1}{2}{x}^4-\frac{1}{6}{x}^6,$ $y_4=1-x^2+\frac{1}{2}{x}^4-\frac{1}{6}{x}^6+\frac{1}{24}{x}^8$; $y(x)=\exp (-x^2)$

5. $y_0=0,y_1=2x,y_2=2x+2x^2,$ $y_3=2x+2x^2+\frac{4}{3}{x}^3,y_4=2x+2x^2+\frac{4}{3}{x}^3+\frac{2}{3}{x}^4$; $y(x)=e^{2x}-1$

7. $y_0=0,y_1=x^2,y_2=x^2+\frac{1}{2}{x}^4,y_3=x^2+\frac{1}{2}{x}^4+\frac{1}{6}{x}^6,y_4=x^2+\frac{1}{2}{x}^4+\frac{1}{6}{x}^6+\frac{1}{24}{x}^8$; $y(x)=\exp (x^2)-1$

9. $y_0=1,y_1=(1+x)+\frac{1}{2}{x}^2,y_2=(1+x+x^2)+\frac{1}{6}{x}^3,$ $y_3=(1+x+x^2+\frac{1}{3}{x}^3)+\frac{1}{24}{x}^4$; $y(x)=2e^x-1-x=1+x+x^2+\frac{1}{3}{x}^3+\cdots$

11. $y_0=1,y_1=1+x,y_2=(1+x+x^2)+\frac{1}{3}{x}^3,y_3=(1+x+x^2+x^3)+\frac{2}{3}{x}^4+\frac{1}{3}{x}^5+\frac{1}{9}{x}^6+\frac{1}{63}{x}^7$; $y(x)=\frac{1}{1-x}=1+x+x^2+x^3+x^4+x^5+\cdots$

12. $y_0=1,y_1=1+\frac{1}{2}x,y_2=1+\frac{1}{2}x+\frac{3}{8}{x}^3+\frac{1}{8}{x}^3+\frac{1}{64}{x}^4,y_3=1+\frac{1}{2}x+\frac{3}{8}{x}^2+\frac{5}{16}{x}^3+\frac{13}{64}{x}^4+\cdots$; $y(x)={(1-x)}^{-1/2}$

13. $\begin{array}{l}\left[\begin{array}{r}{x}_0\\ y_0\end{array}\right]=\left[\begin{array}{r}1\\ -1\end{array}\right],\left[\begin{array}{r}{x}_1\\ y_1\end{array}\right]=\left[\begin{array}{r}1+3t\\ -1+3t\end{array}\right],\\ \left[\begin{array}{r}{x}_2\\ y_2\end{array}\right]=\left[\begin{array}{r}1+3t+\frac{1}{2}{t}^2\\ -1+5t-\frac{1}{2}{t}^2\end{array}\right],\\ \left[\begin{array}{r}{x}_3\\ y_3\end{array}\right]=\left[\begin{array}{r}1+3t+\frac{1}{2}{t}^2+\frac{1}{3}{t}^3\\ -1+5t-\frac{1}{2}{t}^2+\frac{5}{6}{t}^3\end{array}\right]\end{array}$

14. $\mathbf{x}(t)=\left[\begin{array}{c}{e}^t+t{e}^t\\ e^t\end{array}\right]$

16. $y_3(1)\approx 0.350185$