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 Yn+1.
But this would NOT be the case for, e.g., y′ = t y3ey (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.