实例：

u'=-3u+6x+5

u(0)=3

解析解：u=2e^(-3x)+2x+1

欧拉法

h=0.01;%步长
x=0:h:1;
u=zeros(length(x),1);
u(1)=3;
for i=1:length(x)-1
    du=-3*u(i)+6*x(i)+5;
    u(i+1)=u(i)+h*du;
end
figure(1)
U=2*exp(-3*x)+2*x+1;
plot(x,U,'r')%解析解
hold on
plot(x,u,'bo','MarkerFaceColor','b')%数值解

改进的欧拉法

clc
clear
h=0.01;%步长
x=0:h:1;
u=zeros(length(x),1);
u(1)=3;
for i=1:length(x)-1
    du=-3*u(i)+6*x(i)+5;
    u_ba=u(i)+h*du;
    du1=-3*u_ba+6*x(i+1)+5;
    u(i+1)=u(i)+h*0.5*(du+du1);
end
hold on
plot(x,u,'k^','MarkerFaceColor','k')

ode45求解

function  dudt=odefun(x,u)
dudt=-3*u+6*x+5;
end

clc
clear
xspan=0:0.01:1;
u0=3;
[x,u] = ode45('odefun', xspan, u0);
hold on
plot(x,u,'bs','MarkerFaceColor','b')

总体

function  dudt=odefun(x,u)
dudt=-3*u+6*x+5;
end

clc
clear

h=0.01;%步长
x=0:h:1;
u=zeros(length(x),1);
u(1)=3;
for i=1:length(x)-1
    du=-3*u(i)+6*x(i)+5;
    u(i+1)=u(i)+h*du;
end
figure(1)
U=2*exp(-3*x)+2*x+1;
plot(x,U,'r')
hold on
plot(x,u,'bo')

for i=1:length(x)-1
    du=-3*u(i)+6*x(i)+5;
    u_ba=u(i)+h*du;
    du1=-3*u_ba+6*x(i+1)+5;
    u(i+1)=u(i)+h*0.5*(du+du1);
end
hold on
plot(x,u,'go')

xspan=0:0.01:1;
u0=3;
[x,u] = ode45('odefun', xspan, u0);
hold on
plot(x,u,'go')
legend('解析解','欧拉法的解','改进欧拉法的解','ode45')