F1.sage

def generator():
    t = var("t")
    y,yp = mi_vars("y","y'")
    t0 = randrange(-2,3)
    y0 = randrange(-2,3)
    field_sage,dydt = choice([
        ("sin(t+y)",sin(t+y)),
        ("cos(t+y)",cos(t+y)),
        ("y/15-t/3",y/15-t/3),
        ("t/3-y/15",t/3-y/15),
        ("sin(y/2)",sin(y/2)),
        ("cos(y/2)",cos(y/2)),
        ("t*y/9-t/3",t*y/9-t/3),
        ("t*y/9+t/3",t*y/9+t/3),
    ])
    dt = choice([-1,1])/4
    tn = t0
    yn = y0
    for _ in range(8):
        dydtn = dydt.subs({t:tn,y:yn}).n(digits=5)
        tn = (tn + dt).n(digits=5)
        yn = (yn + dydtn*dt).n(digits=5)

    return {
        "field_sage": field_sage,
        "ode": (yp==dydt),
        "y0": y0,
        "t0": t0,
        "othert": t0+8*dt,
        "othery": yn.n(digits=2),
    }