请用matlab代码完成微分方程数值解： 初值问题y'=-50y+50x^2+2x,0≤x≤1, y(0)=1/3 (1)采用Euler法求解初值问题，包括向前欧拉，向后欧拉，梯形法和改进欧拉法，区间数目n=100. (2)采用Runge-Kutta法求解初值问题，包括2阶，3阶和4阶龙格-库塔法，区间数目n=100.

好的，下面是针对该初值问题的matlab代码实现，包括Euler法和Runge-Kutta法的实现，其中，Euler法包括向前欧拉、向后欧拉、梯形法和改进欧拉法，Runge-Kutta法包括2阶、3阶和4阶龙格-库塔法：

Euler法：

% 向前欧拉法
function y = forward_euler(f, h, x0, y0, xn)
    x = x0:h:xn;
    y = zeros(size(x));
    y(1) = y0;
    for i = 1:length(x)-1
        y(i+1) = y(i) + h*f(x(i), y(i));
    end
end

% 向后欧拉法
function y = backward_euler(f, h, x0, y0, xn)
    x = x0:h:xn;
    y = zeros(size(x));
    y(1) = y0;
    for i = 1:length(x)-1
        phi = @(y) y - h*f(x(i+1), y) - y(i);
        y(i+1) = fzero(phi, y(i));
    end
end

% 梯形法
function y = trapezoidal(f, h, x0, y0, xn)
    x = x0:h:xn;
    y = zeros(size(x));
    y(1) = y0;
    for i = 1:length(x)-1
        phi = @(y) y - y(i) - 0.5*h*(f(x(i), y(i)) + f(x(i+1), y));
        y(i+1) = fzero(phi, y(i));
    end
end

% 改进欧拉法
function y = improved_euler(f, h, x0, y0, xn)
    x = x0:h:xn;
    y = zeros(size(x));
    y(1) = y0;
    for i = 1:length(x)-1
        y_star = y(i) + h*f(x(i), y(i));
        y(i+1) = y(i) + 0.5*h*(f(x(i), y(i)) + f(x(i+1), y_star));
    end
end

% 测试
f = @(x, y) -50*y + 50*x^2 + 2*x;
h = 1/100;
x0 = 0;
y0 = 1/3;
xn = 1;
y_forward_euler = forward_euler(f, h, x0, y0, xn);
y_backward_euler = backward_euler(f, h, x0, y0, xn);
y_trapezoidal = trapezoidal(f, h, x0, y0, xn);
y_improved_euler = improved_euler(f, h, x0, y0, xn);

x = 0:h:xn;
y_exact = exp(-50*x) + x.^2/3 + (2/25)*x - (1/75);
plot(x, y_forward_euler, 'r-', x, y_backward_euler, 'g-', x, y_trapezoidal, 'b-', x, y_improved_euler, 'm-', x, y_exact, 'k-');
legend('Forward Euler', 'Backward Euler', 'Trapezoidal', 'Improved Euler', 'Exact');
xlabel('x');
ylabel('y');
title('Solutions of ODE using Euler methods');

Runge-Kutta法：

% 2阶龙格-库塔法
function y = rk2(f, h, x0, y0, xn)
    x = x0:h:xn;
    y = zeros(size(x));
    y(1) = y0;
    for i = 1:length(x)-1
        k1 = h*f(x(i), y(i));
        k2 = h*f(x(i)+h, y(i)+k1);
        y(i+1) = y(i) + 0.5*(k1 + k2);
    end
end

% 3阶龙格-库塔法
function y = rk3(f, h, x0, y0, xn)
    x = x0:h:xn;
    y = zeros(size(x));
    y(1) = y0;
    for i = 1:length(x)-1
        k1 = h*f(x(i), y(i));
        k2 = h*f(x(i)+0.5*h, y(i)+0.5*k1);
        k3 = h*f(x(i)+h, y(i)-k1+2*k2);
        y(i+1) = y(i) + (1/6)*(k1+4*k2+k3);
    end
end

% 4阶龙格-库塔法
function y = rk4(f, h, x0, y0, xn)
    x = x0:h:xn;
    y = zeros(size(x));
    y(1) = y0;
    for i = 1:length(x)-1
        k1 = h*f(x(i), y(i));
        k2 = h*f(x(i)+0.5*h, y(i)+0.5*k1);
        k3 = h*f(x(i)+0.5*h, y(i)+0.5*k2);
        k4 = h*f(x(i)+h, y(i)+k3);
        y(i+1) = y(i) + (1/6)*(k1+2*k2+2*k3+k4);
    end
end

% 测试
f = @(x, y) -50*y + 50*x^2 + 2*x;
h = 1/100;
x0 = 0;
y0 = 1/3;
xn = 1;
y_rk2 = rk2(f, h, x0, y0, xn);
y_rk3 = rk3(f, h, x0, y0, xn);
y_rk4 = rk4(f, h, x0, y0, xn);

x = 0:h:xn;
y_exact = exp(-50*x) + x.^2/3 + (2/25)*x - (1/75);
plot(x, y_rk2, 'r-', x, y_rk3, 'g-', x, y_rk4, 'b-', x, y_exact, 'k-');
legend('RK2', 'RK3', 'RK4', 'Exact');
xlabel('x');
ylabel('y');
title('Solutions of ODE using Runge-Kutta methods');