def eluer(rangee,h,fun,x0,y0):
    step = int(rangee/h)
    x = [x0] + [h * i for i in range(step)]
    u = [y0] + [0     for i in range(step)]
    for i in range(step):
        u[i+1] = u[i] + h * fun(x[i],u[i])
    plt.plot(x,u,label = "eluer")
    return u

def implicit_euler(rangee,h,fun,x0,y0):
    step = int(rangee/h)
    x = [x0] + [h * i for i in range(step)]
    u = [y0] + [0     for i in range(step)]
    v = ["null"] + [0 for i in range(step)]
    for i in range(step):
            v[i+1] = u[i] + h * fun(x[i],u[i])
            u[i+1] = u[i] + h/2 * (fun(x[i],u[i]) + fun(x[i],v[i+1]))
    plt.plot(x,u,label = "implicit eluer")
    return u

def order_3_runge_kutta(rangee,h,fun,x0,y0):
    step = int(rangee/h)
    k1,k2,k3 = [[0 for i in range(step)] for i in range(3)]
    x = [x0] + [h * i for i in range(step)]
    y = [y0] + [0     for i in range(step)]
    for i in range(step):
        k1[i] = fun(x[i],y[i])
        k2[i] = fun(x[i]+0.5*h,y[i]+0.5*h*k1[i])
        k3[i] = fun(x[i]+0.5*h,y[i]+2*h*k2[i]-h*k1[i])
        y[i+1] = y[i] + 1/6 * h * (k1[i]+4*k2[i]+k3[i])
    plt.plot(x,y,label = "order_3_runge_kutta")
    return y

def order_4_runge_kutta(rangee,h,fun,x0,y0):
    step = int(rangee/h)
    k1,k2,k3,k4 = [[0 for i in range(step)] for i in range(4)]
    x = [x0] + [h * i for i in range(step)]
    y = [y0] + [0     for i in range(step)]
    for i in range(step):
        k1[i] = fun(x[i],y[i])
        k2[i] = fun(x[i]+0.5*h,y[i]+0.5*h*k1[i])
        k3[i] = fun(x[i]+0.5*h,y[i]+0.5*h*k2[i])
        k4[i] = fun(x[i]+h,y[i]+h*k3[i])
        y[i+1] = y[i] + 1/6 * h * (k1[i]+2*k2[i]+2*k3[i]+k4[i])
    plt.plot(x,y,label = "order_4_runge_kutta")
    return y

rangee = 1
fun = lambda x,y:y-2*x/y
 
implicit_euler(rangee,0.0001,fun,0,1)
order_4_runge_kutta(rangee,0.0001,fun,0,1)
order_3_runge_kutta(rangee,0.0001,fun,0,1)
eluer(rangee,0.0001,fun,0,1)
plt.legend()
plt.show()