Formula for the exact solution u1(u) and u2(t)?

I want to examine the system u´(t)=Au(t) for matrix A=[12 15; -10 -13] and startvalue u0=[3; -2]

I plotted the system with code:

A=[12 15;-10 -13];

u1=linspace(-3,2,20);u2=linspace(-3,2,20);

[U1,U2]=meshgrid(u1,u2);

F1=A(1,1)*U1+A(1,2)*U2;

F2=A(2,1)*U1+A(2,2)*U2;

quiver(U1,U2,F1,F2,1.5)

axis([-3 5 -3 3]), hold on

f=@(t,u)A*u;

u0=[3;-2];

[t,U]=ode45(f,[0 5],u0);

plot(U(:,1),U(:,2),'r','LineWidth',2)

but how do I find the formula for the exact solution u1(u) and u2(t)?

Answer 1

[Stephan]

Hi,

try:

syms u1(t) u2(t)
A = [12 15; -10 -13];
cond = [u1(0) == 3; u2(0) == -2]
ode = [diff(u1,t); diff(u2,t)] == A * [u1; u2]
result = dsolve(ode, cond)
result_u1 = result.u1
result_u2 = result.u2

results are:

result_u1 =
 
3*exp(2*t)
 
 
result_u2 =
 
-2*exp(2*t)

Best regards

Stephan