Consider the initial value problem: $y(0) = 1, y ′ (t) = λy(t)$
Using that the solution is $y(t) = e^{λt}$, write out a Taylor series for $y(t_{i+1})$ about $y(t_i)$ up to terms of order $h^4$ (note: use that $h = t_{i+1} − t_i$).
Write out what the RK32 method would be for this problem. Show that it agrees with the Taylor Series expansion up to terms of order $h^3$.
I have determined the Taylor series for $y(t_{i+1})$ about $y(t_i)$ as
$y(t_i) + \lambda y(t_i)h + \frac{1}{2}\lambda^2 y(t_i)h^2 + \frac{1}{3!}\lambda^3 y(t_i)h^3 + \cdots$
But I am not able to write out the equation in RK3-2 form
RK3-2 is defined as
$k_1 = f(t_i, y_i)$
$k_2 = f(t_i + \frac{1}{2}h, y_i + \frac{1}{2}hK_1)$
$k_3 = f(t_i + h, y_i - hk_1 + 2hK_2)$
$y_{i+1} = y_i + h(\frac{1}{6}k_1 + \frac{4}{6}k_2 +\frac{1}{6}k_3)$
How can I show that the RK3-2 method agrees with the Taylor expansion up to terms of order $h^3$?