Question

Q. 6 Homogenous Linear Systems Using the notion of eigenvalues and eigenvectors, solve the given system of differential equations. a. Distinct Eigenvalues $\frac{dx}{dt} = 2x + 3y$, $\frac{dy}{dt} = 2x + y$. b. Repeated Eigenvalues (Two Cases) First Case. $x' = \begin{bmatrix} 1 & -2 & 2\\ -2 & 1 & -2\\ 2 & -2 & 1 \end{bmatrix} x$. Second Case (An Eigenvector is lost) $x' = \begin{bmatrix} 3 & -18\\ 2 & -9 \end{bmatrix} x$. c. Complex Eigenvalues $\frac{dx}{dt} = 6x - y$, $\frac{dy}{dt} = 5x + 4y$.

          Q. 6 Homogenous Linear Systems
Using the notion of eigenvalues and eigenvectors, solve the given system of differential equations.
a. Distinct Eigenvalues
$\frac{dx}{dt} = 2x + 3y$, $\frac{dy}{dt} = 2x + y$.
b. Repeated Eigenvalues (Two Cases)
First Case.
$x' = \begin{bmatrix} 1 & -2 & 2\\ -2 & 1 & -2\\ 2 & -2 & 1 \end{bmatrix} x$.
Second Case (An Eigenvector is lost)
$x' = \begin{bmatrix} 3 & -18\\ 2 & -9 \end{bmatrix} x$.
c. Complex Eigenvalues
$\frac{dx}{dt} = 6x - y$, $\frac{dy}{dt} = 5x + 4y$.

Q. 6 Homogenous Linear Systems
Using the notion of eigenvalues and eigenvectors, solve the given system of differential equations.
a. Distinct Eigenvalues
(dx)/(dt) = 2x + 3y, (dy)/(dt) = 2x + y.
b. Repeated Eigenvalues (Two Cases)
First Case.
x' =
< b m a t r i x >
x.
Second Case (An Eigenvector is lost)
x' =
< b m a t r i x >
x.
c. Complex Eigenvalues
(dx)/(dt) = 6x - y, (dy)/(dt) = 5x + 4y.

Added by Jesse P.

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