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I want to solve the following difference equation for all $0<h<1$ and $c>0$: $$\frac{a_{k+1}-a_k}{h}=-a_k $$ with the condition $a_0=c$. Can someone explain how to solve equations of this kind?

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    I suggest you solve for $a_{k+1}$, do a few iterations, and see if you can't find a pattern. Have you tried anything? – Nick F Nov 29 '22 at 22:11

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Hint: Rewrite it as $a_{k+1} = (1-h)a_k = (1-h)^2a_{k-1}=...$. Can you take it from here?

Wang YeFei
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