Using Euler's method (the method of eigenvalues and eigenvectors, the case of different complex eigenvalues) solve the sytem of differential equations $\overrightarrow{y'}$ = A$\overrightarrow{y}$ or solve the initial value problem $\overrightarrow{y'}$= A$\overrightarrow{y}$, $\overrightarrow{y}$(0) = $\overrightarrow{y}_0$, if
\begin{bmatrix}-7&1\\-2&−5\end{bmatrix}
My attempt: (but unfortunately it is not Euler's method, i don't understand this method):
So i have started from forming a new matrix by subtracting $λ$ from the diagonal entries of the given matrix:\begin{bmatrix}−λ-7&1\\-2&−λ−5\end{bmatrix} The determinant of the obtained matrix is $λ_2+12λ+37$
Solve the equation $λ^2+12λ+37=0.$
The roots are $λ1=−6−i, λ2=−6+i$, these are the eigenvalues.
Next, i have found the eigenvectors.
- $λ=−6−i$
$\begin{bmatrix}−λ-7&1\\-2&−λ−5\end{bmatrix} = \begin{bmatrix}−1+i&1\\-2&i+1\end{bmatrix}$
The null space of this matrix is $\begin{bmatrix}1/2+i/2\\ 1\end{bmatrix}$,
This is the eigenvector.
- $λ=−6+i$
$\begin{bmatrix}−λ-7&1\\-2&−λ−5\end{bmatrix} = \begin{bmatrix}−1+i&1\\-2&i-1\end{bmatrix}$,
The null space of this matrix is $\begin{bmatrix}1/2-i/2\\ 1\end{bmatrix}$,
and this is the eigenvector.
And i dont know what i should do next to calculate the fundamental equations (formula) or any other solution is of the following form.
For every pair of complex eigenvalues $p\pm qi$ your solutions get a component of the form $e^{px}(a\cos qx + b\sin qx)$. In your case they look as: $y_{k}=e^{-6x}(a_k\cos x + b_k\sin x)$ where $k\in (1,2)$. The two sets of constants $a_k,b_k$ can be figured out from the system's matrix and the boundary conditions.