\text { If } \mathbf{A} \underline{\mathbf{x}}=\lambda \underline{\mathbf{x}} \text {, determine the eigenvalues and the corresponding eigenvectors for } A=\left(\begin{array}{ll} 1 & 4 \\ 2 & 3 \end{array}\right) Hence, write down

the associated modal matrix P and diagonal matrix_D,and use these values to solve the following system differential equations: \dot{x}_{1}=x_{1}+4 x_{2} \dot{\mathrm{x}}_{2}=2 \mathrm{x}_{1}+3 \mathrm{x}_{2} \text { Given that when } t=0, x_{1}=0 \text { and } x_{2}=2 \text {. } \text { Use the } 4^{\text {th }} \text { - order Runge Kutta method to solve the differential equation: } \frac{d y}{d x}=e^{x} y for values of x = 0 (0.2) 0.4 given that y = 1 when x = 0. Give your answers correct to 5 decimal places. Obtain the analytical solution of the differential equation and compare the analytical solution when x = 0.4 with the values obtained using Runge-Kutta.