MT, P5

[Victor Ivrii]

Apply the Euler method with step $h=\frac{1}{3}$ to the initial value problem
\begin{equation*}
y'=3y^2-t;\quad y(0)=0
\end{equation*}
to find approximate values of $y(\frac{1}{3}), y(\frac{2}{3})$ and $y(1)$.

[Xuewen Yang]

see attach

[Xiaozeng Yu]

5

[Huan Ying Huang]

$y(0)=0$,
$$y_n=y_{n-1}+h f(t_n,y_n),$$
with $f(t,y)=3y^2-t$.

$t_0=0$, $y_0=0$,  $f(t_0,y_0)=0$,

$y_1=0+\frac{1}{3}\cdot 0=0$.

$t_1=\frac{1}{3}$,   $y_1=0$,  $f(t_1,y_1)=-\frac{1}{3}$,

$y_2= 0+ \frac{1}{3}\cdot (-\frac{1}{3})=-\frac{1}{9}$,

$t_2=\frac{2}{3}$,   $y_2=-\frac{1}{9}$,  $f(t_2,y_2)=-\frac{17}{27}$,

$y_3=-\frac{1}{9}+\frac{1}{3}\cdot (-\frac{17}{27}) =-\frac{26}{81}$.

I edited this post. Do not use "x" or "*" for multiplication (* is a convolution, to be studied in APM346 f.e.)  V.I.