2.4

1.5

2.0

def rk3(f, y0, t0, tf, h):

    t_values = np.arange(t0, tf + h, h)

    y_values = np.zeros_like(t_values)

    y_values[0] = y0

    for i in range(1, len(t_values)):

        k1 = f(t_values[i-1], y_values[i-1])

        k2 = f(t_values[i-1] + h/2, y_values[i-1] + h*k1/2)

        k3 = f(t_values[i-1] + h, y_values[i-1] - 2*h*k1 + h*k2)

        y_values[i] = y_values[i-1] + h*(k1 + 2*k2 + k3)/6

    return t_values, y_values

def rk3(f, y0, t0, tf, h):

    t_values = np.arange(t0, tf + h, h)

    y_values = np.zeros_like(t_values)

    y_values[0] = y0

    for i in range(1, len(t_values)):

        k1 = f(t_values[i-1], y_values[i-1])

        k2 = f(t_values[i-1] + h/2, y_values[i-1] + h*k1/2)

        k3 = f(t_values[i-1] + h, y_values[i-1] - h*k1 + 2*h*k2)

        y_values[i] = y_values[i-1] + h*(k1 + 4*k2 + k3)/6

    return t_values, y_values

def rk3(f, y0, t0, tf, h):

    t_values = np.arange(t0, tf + h, h)

    y_values = np.zeros_like(t_values)

    y_values[0] = y0

    for i in range(1, len(t_values)):

        k1 = f(t_values[i-1], y_values[i-1])

        k2 = f(t_values[i-1] + h, y_values[i-1] + h*k1)

        k3 = f(t_values[i-1] + h, y_values[i-1] + h*k2)

        y_values[i] = y_values[i-1] + h*(k1 + k2 + k3)/3

    return t_values, y_values

def rk3(f, y0, t0, tf, h):

    t_values = np.arange(t0, tf + h, h)

    y_values = np.zeros_like(t_values)

    y_values[0] = y0

    for i in range(1, len(t_values)):

        k1 = f(t_values[i-1], y_values[i-1])

        k2 = f(t_values[i-1] + h/3, y_values[i-1] + h*k1/3)

        k3 = f(t_values[i-1] + 2*h/3, y_values[i-1] + 2*h*k2/3)

        y_values[i] = y_values[i-1] + h*(k1 + 3*k2 + 3*k3)/7

    return t_values, y_values

def rk3(f, y0, t0, tf, h):

    t_values = np.arange(t0, tf + h, h)

    y_values = np.zeros_like(t_values)

    y_values[0] = y0

    for i in range(1, len(t_values)):

        k1 = f(t_values[i-1], y_values[i-1])

        k2 = f(t_values[i-1] + h/2, y_values[i-1] + h*k1/2)

        k3 = f(t_values[i-1] + h, y_values[i-1] + h*k2)

        y_values[i] = y_values[i-1] + h*(k1 + 4*k2 + k3)/6

    return t_values, y_values

def modified_euler(f, x0, y0, h, steps):

    for _ in range(steps):

        y_pred = y0 + h * f(x0, y0)

        y0 += 0.5 * h * (f(x0, y0) + f(x0 + h/2, y+y_pred/2))

        x0 += h

    return y0

def modified_euler(f, x0, y0, h, steps):

    for _ in range(steps):

        y_pred = y0 + h * f(x0, y0)

        y0 += 0.5 * h * (f(x0, y0) + f(x0 + h, y_pred))

        x0 += h

    return y0

def modified_euler(f, x0, y0, h, steps):

    for _ in range(steps):

        y_pred = y0 + h * f(x0, y0)

        y0 += h * f(x0 + h, y_pred)

        x0 += h

    return y0

def modified_euler(f, x0, y0, h, steps):

    for _ in range(steps):

        y_pred = y0 + h * f(x0, y0)

        y0 += 0.5 * h * (f(x0, y0) + f(x0 + h/2, y_pred))

        x0 += h

    return y0

def modified_euler(f, x0, y0, h, steps):

    for _ in range(steps):

        y_pred = y0 + h * f(x0, y0)

        y0 += 0.5 * h * (f(x0, y0) + f(x0 + h/2, y+y_pred))

        x0 += h

    return y0

((x-1)(x-2))/((4-1)(4-2))

((x-2)(x-4))/((1-2)(1-4))

    def runge_kutta_4th_order(f, x0, y0, h, n):

    x = x0

    y = y0

    for _ in range(n):

        k1 = h * f(x, y)

        k2 = h * f(x + h/2, y + k1/2)

        k3 = h * f(x + h/2, y + k2/2)

        k4 = h * f(x + h, y + k3)

        y = y + (k1 + k2 + k3 + k4) / 4

        x = x + h

    return y

    def runge_kutta_4th_order(f, x0, y0, h, n):

    x = x0

    y = y0

    for _ in range(n):

        k1 = h * f(x, y)

        k2 = h * f(x + h/2, y + k1/2)

        k3 = h * f(x + h/2, y + k2/2)

        k4 = h * f(x + h, y + k3)

        y = y + (k1 + 2*k2 + 2*k3 + k4) / 6

        x = x + h

    return y

   def runge_kutta_4th_order(f, x0, y0, h, n):

    x = x0

    y = y0

    for _ in range(n):

        k1 = h * f(x, y)

        k2 = h * f(x + h, y + k1)

        k3 = h * f(x + h, y + k2)

        k4 = h * f(x + h, y + k3)

        y = y + (k1 + k2 + k3 + k4) / 4

        x = x + h

    return y

    def runge_kutta_4th_order(f, x0, y0, h, n):

    x = x0

    y = y0

    for _ in range(n):

        k1 = h * f(x, y)

        k2 = h * f(x + h, y + k1)

        k3 = h * f(x + h, y + k2)

        k4 = h * f(x + h, y + k3)

        y = y + (k1 + 2*k2 + 2*k3 + k4) / 6

        x = x + h

    return y

    def runge_kutta_4th_order(f, x0, y0, h, n):

    x = x0

    y = y0

    for _ in range(n):

        k1 = h * f(x, y)

        k2 = h * f(x + h/2, y + k1)

        k3 = h * f(x + h/2, y + k2)

        k4 = h * f(x + h, y + k3)

        y = y + (k1 + 2*k2 + 2*k3 + k4) / 6

        x = x + h

    return y

1.5

def lagrange_interpolation(x, y, x_val):

    result = 0

    for i in range(len(x)):

        term = y[i]

        for j in range(len(x)):

            if i != j:

                term /= (x[i] - x[j])

        result += term

    return result

def lagrange_interpolation(x, y, x_val):

    result = 0

    for i in range(len(x)):

        term = y[i]

        for j in range(len(x)):

            term *= (x_val - x[j]) / (x[i] - x[j])

        result += term

    return result

   def lagrange_interpolation(x, y, x_val):

    result = 0

    for i in range(len(x)):

        term = y[i]

        for j in range(len(x)):

            if i != j:

                term *= (x_val - x[i]) / (x[j] - x[i])

        result += term

    return result

def lagrange_interpolation(x, y, x_val):

    result = 0

    for i in range(len(x)):

        term = y[i]

        for j in range(len(x)):

            if i != j:

                term *= (x_val - x[j]) / (x[i] - x[j])

        result += term

    return result

def lagrange_interpolation(x, y, x_val):

    result = 0

    for i in range(len(x)):

        term = y[i]

        for j in range(len(x)):

            if i == j:

                term *= (x_val - x[j]) / (x[i] - x[j])

        result += term

    return result