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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!

The exercise in question.

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    $\begingroup$ 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 $\endgroup$ – nicoguaro May 24 at 13:59
  • $\begingroup$ @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. $\endgroup$ – Logarithmic Derivative May 24 at 14:06

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