Chapter 8.2, Problem 14P

Using three terms in the Taylor series given in Eq. (12)and taking $h=0.1$, determine approximate values of the solution of the illustrative example $\begin{array}{cc}y\text{'}=1-t+4y& y(0)=1\end{array}$ at $t=0.1$ and $0.2$. Compare the results with those using theEuler method and with the exact values.

Hint: If $y\text{'}=f(t,y),$ what is $y\text{'}\text{'}$ ?

$\varphi \left(t_{n+1}\right)=\varphi \left(t_n\right)+f\left[t_n,\varphi \left(t_n\right)\right]h+\varphi \text{'}\text{'}\left(t_n\right)\frac{h^2}{2!}+\cdots$. (12)