CC3.sage

def generator():
    t = var("t")
    c1 = var("c_1")
    c2 = var("c_2")
    i = var('i')
    k1,k2,k3 = var("k_1 k_2 k_3")
    y,yp,ypp,yppp = mi_vars("y","y'","y''","y'''")
    odes = []

    # pick a,b for (D-a)^2y+b^2=y''-2ay'+a^2+b^2=0
    a = randrange(1,6)*choice([-1,1])
    b = randrange(1,6)
    ode = (shuffled_equation(ypp,-2*a*yp,(a^2+b^2)*y)*randrange(2,4))
    ode_sol = (y==exp(a*t)*(k1*cos(b*t)+k2*sin(b*t)))
    complex_sol = (y==c1*e^(a*t+b*i*t)+c2*e^(a*t-b*i*t))
    odes.append({"ode":ode,"sol":ode_sol,"complex":complex_sol})

    # pick c for (D-c)^2y=0
    c = randrange(1,11)*choice([-1,1])
    ode = (shuffled_equation(ypp,-2*c*yp,c^2*y)*randrange(2,4))
    ode_sol = (y==k2*exp(c*t)+k1*t*exp(c*t))
    odes.append({"ode":ode,"sol":ode_sol})

    # pick a,b for D(D-a)(D-b)y=0
    a = randrange(1,6)*choice([-1,1])
    b=a
    while b==a:
        b = randrange(1,6)*choice([-1,1])
    ode = (shuffled_equation(yppp,(-a-b)*ypp,a*b*yp)*randrange(2,4))
    ode_sol = (y==k2*exp(a*t)+k1*exp(b*t)+k3)
    odes.append({"ode":ode,"sol":ode_sol})

    shuffle(odes)

    return {
        "odes": odes,
    }