Suppose that an m-stage ERK of order m is applied to the linear scalar equation $y'=\lambda y $. Prove that $$y_n=[\sum^{m}_{j=1}1/k! (h\lambda)^k]^ny_0$$
I know that the exact solution resembles the RHS of the equation. But I am unsure of how the nth power comes into play and how I am to use the ERK to get it. The scheme for the ERK is $y_{n+1}=y_n+h\sum^{s}_{j=1}b_jf(t_n+c_jh,z_j)$ which outputs $y_{n+1}$ and not $y_n$. Any assistance on this is appreciated.
So I can use a recursive scenario to produce the nth power. So if,
$$y_{n+1}=[\sum^{m}_{k=0}1/k!(h\lambda)^k]y_n$$ then$$y_1=[\sum^{m}_{k=0}1/k!(h\lambda)^k]y_0$$ $$y_2=[\sum^{m}_{k=0}1/k!(h\lambda)^k]^2y_0$$and so on so that $$y_n=[\sum^{m}_{k=0}1/k!(h\lambda)^k]^ny_0$$