Systems of DE (Eigenvectors)

Question

In a $2\times 2$ system of differential equations with initial conditions how are the unknowns in the general solution ($C_1$ and $C_2$) calculated given that there are an infinite number of eigenvectors possible? Won't that fact, which i'm assuming is true, yield infinitely many solutions, all affecting how the solution trajectories look?

Thanks.

Answer 1

The general form solution should be (with real eigenvalues) $$\vec{x}(t)=c_1e^{\lambda_1t}\vec{\eta}_1+c_2e^{\lambda_2t}\vec{\eta}_2$$ where $\lambda$ are the eigenvalues and $\vec{\eta}$ are the eigenvectors. The arbitrary constants $c_1$ and $c_2$ are there because, as you say, there are an infinite amount of eigenvectors that satisfy the system, which is why we need the initial conditions.

Answer 2

In order to find the general solution you only need two linearly independent eigenvectors along with the eigenvalues. The general solution is $$y= C_1 e^{\lambda _1t}V_1 + C_2 e^{\lambda _2t}V_2$$ where the coefficients are found from the initial values.