在matlab上y'=2*y./x + x.^2.*exp(x)用四阶标准龙格—库塔法、步长h = 0.1解数值解

可以按照以下步骤在 Matlab 中使用四阶标准龙格-库塔法求解该微分方程的数值解：

1. 定义微分方程

function dydx = myode(x, y)
dydx = 2*y./x + x.^2.*exp(x);
end

2. 定义步长和求解区间

h = 0.1;
xspan = [1 2];

3. 初始化初值条件

y0 = 1;
x0 = xspan(1);

4. 使用龙格-库塔方法求解微分方程

[x, y] = ode4(@myode, xspan, y0, h);

其中 `ode4` 是一个自定义函数，实现龙格-库塔方法的求解过程。其代码如下：

function [x, y] = ode4(odefun, xspan, y0, h)
x0 = xspan(1);
xf = xspan(2);
x = x0:h:xf;
y = zeros(size(x));
y(1) = y0;
for n = 1:length(x)-1
    k1 = odefun(x(n), y(n));
    k2 = odefun(x(n)+h/2, y(n)+h/2*k1);
    k3 = odefun(x(n)+h/2, y(n)+h/2*k2);
    k4 = odefun(x(n)+h, y(n)+h*k3);
    y(n+1) = y(n) + h/6*(k1 + 2*k2 + 2*k3 + k4);
end
end

5. 可视化数值解

plot(x, y, '-o')
xlabel('x')
ylabel('y')

完整代码如下：

function dydx = myode(x, y)
dydx = 2*y./x + x.^2.*exp(x);
end

function [x, y] = ode4(odefun, xspan, y0, h)
x0 = xspan(1);
xf = xspan(2);
x = x0:h:xf;
y = zeros(size(x));
y(1) = y0;
for n = 1:length(x)-1
    k1 = odefun(x(n), y(n));
    k2 = odefun(x(n)+h/2, y(n)+h/2*k1);
    k3 = odefun(x(n)+h/2, y(n)+h/2*k2);
    k4 = odefun(x(n)+h, y(n)+h*k3);
    y(n+1) = y(n) + h/6*(k1 + 2*k2 + 2*k3 + k4);
end
end

h = 0.1;
xspan = [1 2];
y0 = 1;
[x, y] = ode4(@myode, xspan, y0, h);
plot(x, y, '-o')
xlabel('x')
ylabel('y')