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A system of differential equations can be created for tWO masses connected by springs between one another; and connected opposing walls_ The dependent variables for...

Question

A system of differential equations can be created for tWO masses connected by springs between one another; and connected opposing walls_ The dependent variables form X [ vector y consisting of the displacement and velocity of each of the two masses. For the system Ay, the matrix 4 is given by:-17Because the system oscillates there will be complex eigenvalues. Find the eigenvalue associated with the following eigenvector:S5i 5i20 + 15i 20 15i

A system of differential equations can be created for tWO masses connected by springs between one another; and connected opposing walls_ The dependent variables form X [ vector y consisting of the displacement and velocity of each of the two masses. For the system Ay, the matrix 4 is given by: -17 Because the system oscillates there will be complex eigenvalues. Find the eigenvalue associated with the following eigenvector: S5i 5i 20 + 15i 20 15i



Answers

Consider the system $\mathbf{x}^{\prime}(t)=\mathbf{A} \mathbf{x}(t), t \geq 0,$ with
$$ \mathbf{A}=\left[ \begin{array}{rr}{-2} & {1} \\ {1} & {-2}\end{array}\right] $$
(a) Show that the matrix $A$ has eigenvalues $r_{1}=-1$ and $r_{2}=-3$ with corresponding eigenvectors $\mathbf{u}_{1}=\operatorname{col}(1,1)$ and $\mathbf{u}_{2}=\operatorname{col}(1,-1)$
(b) Sketch the trajectory of the solution having initial vector $\mathbf{x}(0)=\mathbf{u}_{1} .$
(c) Sketch the trajectory of the solution having initial
vector $\mathbf{x}(0)=-\mathbf{u}_{2}$
(d) Sketch the trajectory of the solution having initial $\quad$ vector $\mathbf{x}(0)=\mathbf{u}_{1}-\mathbf{u}_{2}$


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