"Whats the result of one step of Backward Euler Method with h = 0.1 applied on the IVP:
$ y'(t) = 5y(t) + 10 $
$ y(0) = 1 $
"?
So, been trying to understand the Backward Euler Method for a while now and almost get it. However, I don't know exactly what to do and would really appreciate to see how someone who knows what they're doing tackle this problem.
Best Answer
The method says to take the slope from the next point, which is the unknown one, which is why this is an implicit method. Fortunately your equation is linear so that solving the implicit step is easy, $$ y_1=y_0+hf(t_1,y_1)=1+0.1\cdot(5y_1+10)=2+0.5y_1. $$ This you can now solve for $y_1$.