91Ó°ÊÓ

Consider the differential equation $$ \frac{d^{2} y}{d t^{2}}+2 c \frac{d y}{d t}+k y=0 $$ where \(c\) and \(k\) are positive constants, that governs the behavior of a spring-mass system. Convert the differential equation to a first-order linear system and sketch the corresponding phase portraits. (You will need to distinguish the three cases \(c^{2}>k, c^{2}

The motion of a certain physical system is described by the system of differential equations $$ x_{1}^{\prime}=x_{2}, \quad x_{2}^{\prime}=-b x_{1}-a x_{2} $$ where \(a\) and \(b\) are positive constants and \(a \neq 2 b .\) Show that the motion of the system dies out as \(t \rightarrow+\infty\)

Use the variation-of-parameters method to determine a particular solution to the nonhomogeneous linear system \(\mathbf{x}^{\prime}=A \mathbf{x}+\mathbf{b} .\) Also find the general solution to the system. \(A=\left[\begin{array}{rrr}2 & -2 & 1 \\ 1 & -4 & 1 \\ 2 & 2 & -3\end{array}\right], \mathbf{b}=\left[\begin{array}{l}t \\ 0 \\\ 1\end{array}\right]\) [Hint: The eigenvalues of \(A \text { are } \lambda=2,-2,-5 .]\)

True or False: If \(X(t)\) is a fundamental matrix for the linear system \(\mathbf{x}^{\prime}=A \mathbf{x},\) then \(X(t)^{T}\) is a fundamental matrix for the linear system \(\mathbf{x}^{\prime}=A^{T} \mathbf{x}\)

Characterize the equilibrium point for the system \(\mathbf{x}^{\prime}=A \mathbf{x}\) and sketch the phase portrait. $$A=\left[\begin{array}{rr} 5 & 9 \\ -2 & -1 \end{array}\right]$$