(Solved): Question \#1 : A second order constant coefficient linear differential equations is given as follow ...

Question \#1 : A second order constant coefficient linear differential equations is given as follows for $t?0$ $\frac{d^{2}x(t)}{d{t}^2}?5\frac{dx(t)}{dt}+6x(t)=0$ with the initial conditions given as $x(0)=1$ and ${\frac{dx(t)}{dt}?}_{t=0}=1$. a) Re-express the given differential equation as a first order differential equation by utilizing matrix and vector notation and in accordance with $\frac{d\mathbf{u}(t)}{dt}=A\mathbf{u}(t)$ form. b) Is the system obtained in (a) stable, neutrally stable of unstable? Determine this using $\mathbf{A}$ matrix. c) Compute the eigenvalues and eigenvectors of $\mathbf{A}$ matrix. d) Using the results computed in (c) find $\mathbf{S}$ and $?$ matrices and show that $\mathbf{A}=\mathbf{S}?{\mathbf{S}}^{?1}$ relationship (i.e., the diagonalization relationship) is a valid relationship. e) Compute $\mathbf{u}(\mathbf{t})$ by utilizing your knowledge gained in previous items and obtain the mathematical expression of $x(t)$ from this vector.