Consider the initial-value problem $\dot y = \sin \pi t$, $y(0) = 0$. Use Euler’s Method with five steps to approximate $y(1)$.

Question

I'm always doubtful about asking if the solution manual is wrong, as that opens me up to the usual embarrassment when someone points out the simple error I have made, however, on we go. [

I had a go and got the answer $0.12$, but I got it in the column under $n=1$, the solution manual has zero for the answer there, then I got the answer $0.31$, but likewise, I got that in the column under $n=2$, not as the answer graphic shows. Additionally, I don't understand why $y_0=1$, when I would have thought it would be $y_0=0$. Can anyone point out the (I suspect) simple error I have made? In case the graphic doesn't show, here is the answer table in raw text:

n   0     1     2     3     4     5
tn  0     0.2   0.4   0.6   0.8   1.0
yn  0.00  0.00  0.12  0.31  0.50  0.62 

Answer 1

You seem to have used the implicit Euler method (or simply misread the indices). As the formula in the image states, in the explicit Euler method you compute $y_{n+1}$ from data at the previous time $t_n$. Thus \begin{align} y_1=y_0+h\sin(\pi t_0)&=0+0.2\sin(\pi\cdot 0)=0 \\ y_2=y_1+h\sin(\pi t_1)&=0+0.2\sin(\pi\cdot 0.2)=0.11755705 \end{align} etc.