# Finding the polynomial for the solution of an ODE

I’m stuck trying to solve part (b) and (c) of the below problem, but part (b) is the one of main concern here as I think (c) should follow easily once (b) is completed.

I don’t know where to start with part (b). My first idea was to try to plug in $$y’=-\lambda y$$ in K1 instead of $$f(t_n,y_n)$$. I didn’t get far with that approach as the second step has the expressions $$t_n+\frac{1}{3}h$$ and $$y_n+\frac{1}{3}K1$$ in the arguments of $$f$$, and I don’t know what would relate to our $$y$$ here. I assume that we would get some expression with different powers of $$\chi=h\lambda$$ so that we could write this as a polynomial of $$\chi$$. I feel like I’m completely lost here, and looking in Applied Numerical Methods by Steven Chapra I don’t find much of use. I kindly ask for any help possible, and gladly provide clarifications. Thanks in advance! • Is this a homework question? If yes, please mention it. I would suggest that you split your question in several posts asking for the different parts – nicoguaro May 24 at 13:59
• @nicoguaro It isn’t a homework question, but rather a question on a past exam. I’m primarily interested in part (b) as I guess that (c) follows from it fairly straightforward. I’ll add that in my description. – Logarithmic Derivative May 24 at 14:06