To get a sense about how numerical schemes for ODEs work,

apply the Forward Euler and the Backward Euler schemes,

BY HAND, to approximate y(2), using Δt=1/2, for:

1.

2.

Note that Backward Euler can be done by hand for these,

only because f(t,y) is linear in y, so can easily solve the equation for
Y_{n+1}.

But this would NOT be the case for,
e.g., y**′** = t y^{3}e^{y} (try one step!).

In general, an equation must be solved numerically at each time step
(using a root finder!).

This is just for gaining insight... NOTHING TO TURN IN.