Matlab解微分方程组

Question

Answer 1

你好，答案仅供参考，一般符号求解和数值计算均可以这样做

syms x1(t) x2(t) 
syms a b real
eq = [diff(x1)==a*x1+b*x2; diff(x2)==-b*x1+a*x2];
conds = [x1(0)==1; x2(0)==1];
[x1,x2] = dsolve(eq,conds);
% (1) 四个解析解
x1_sol1 = subs(x1, [a,b], [0.1,1])
x2_sol1 = subs(x2, [a,b], [0.1,1])
x1_sol2 = subs(x1, [a,b], [0.1,-1])
x2_sol2 = subs(x2, [a,b], [0.1,-1])
x1_sol3 = subs(x1, [a,b], [-0.1,1])
x2_sol3 = subs(x2, [a,b], [-0.1,1])
x1_sol4 = subs(x1, [a,b], [-0.1,-1])
x2_sol4 = subs(x2, [a,b], [-0.1,-1])
% （2）数值解：
a_arr = [0.1, -0.1];
b_arr = [1,-1];
count = 0;
figure(1);clf;
for i = 1:1:numel(a_arr)
    for j = 1:1:numel(b_arr)
        a = a_arr(i); b = b_arr(j);
        count = count + 1;
        subplot(2,2, count)
        odefun = @(t,x) [a*x(1)+b*x(2); -b*x(1)+a*x(2)];
        [t,x] = ode45(odefun, [0, 10*pi], [1;1]);
        plot(x(:,1),x(:,2));%画相图
        title(['a=',num2str(a),',b=',num2str(b)])
    end
end

最后表达式

x1_sol1 =
 
exp(t/10)*cos(t) + exp(t/10)*sin(t)
 
 
x2_sol1 =
 
exp(t/10)*cos(t) - exp(t/10)*sin(t)
 
 
x1_sol2 =
 
exp(t/10)*cos(t) - exp(t/10)*sin(t)
 
 
x2_sol2 =
 
exp(t/10)*cos(t) + exp(t/10)*sin(t)
 
 
x1_sol3 =
 
exp(-t/10)*cos(t) + exp(-t/10)*sin(t)
 
 
x2_sol3 =
 
exp(-t/10)*cos(t) - exp(-t/10)*sin(t)
 
 
x1_sol4 =
 
exp(-t/10)*cos(t) - exp(-t/10)*sin(t)
 
 
x2_sol4 =
 
exp(-t/10)*cos(t) + exp(-t/10)*sin(t)

相图

可知
最终相图都是螺旋进入原点，且ab的正负决定了相图螺旋的方向