Mat 17 C — Luis Rademacher Midterm 2 — May 22 — 50 Minutes

Question

Mat 17 C — Luis Rademacher Midterm 2 — May 22nd — 50 minutes Open book and open notes but every student must write his or her own solution Solve the following system of differential equations, analyze stability of equilibria, and address network IP subnetting and topology design. This assignment includes six mathematical and six networking tasks. You are required to provide detailed solutions, including justifications, proofs, and diagrams where applicable. The mathematical problems involve solving systems of ordinary differential equations (ODEs), analyzing their equilibria and stability via linearization, and plotting isoclines. The networking tasks require subnetting of specified IP address ranges, assigning network addresses to VLANs, loopback interfaces, and WAN links, and designing network topology diagrams using Microsoft Visio.

Dr. Jack HW Helper · Accepted Answer

Solving System of ODEs and Stability Analysis 1. Solving the System of ODEs: The given system is: $$ \frac{dx_1}{dt} = 3x_1 - 2x_2, \quad \frac{dx_2}{dt} = x_1 + 4x_2, \quad \text{with initial conditions } x_1(0)=1, x_2(0)=1. $$ Expressed in matrix form: $$ \frac{d}{dt} \begin{bmatrix} x_1 \ x_2 \end{bmatrix} = \begin{bmatrix} 3 & -2 \ 1 & 4 \end{bmatrix} \begin{bmatrix} x_1 \ x_2 \end{bmatrix}. $$ The coefficient matrix $$ A = \begin{bmatrix} 3 & -2 \ 1 & 4 \end{bmatrix} $$ has characteristic equation: $$ \det(A - \lambda I) = (3-\lambda)(4-\lambda) - (-2)(1) = (\lambda^2 - 7\lambda + 14) = 0. $$ Discriminant $ D = (-7)^2 - 4 \times 1 \times 14 = 49 - 56 = -7 \[ \lambda_{1,2} = \frac{7 \pm i \sqrt{7}}{2}. \] Since the real part \(\frac{7}{2} > 0$, solutions exhibit exponential growth with oscillation, indicating an unstable focus at the origin. 2. Compartment Diagram: The system can be interpreted as a flow between compartments $x_1$ and $x_2$. For the general form: $$ \frac{dx_1}{dt} = a x_1 + b x_2, \quad \frac{dx_2}{dt} = c x_1 + d x_2, $$ the corresponding diagram has arrows from $x_1$ to $x_1$ and $x_2$ with rates $a$ and $b$, respectively, and likewise for $x_2$. Determining $a, b, c, d$ directly from the coefficients yields the compartment flow structure; here, for the given matrix, the diagram shows flow from $x_1$ to itself and $x_2$, and vice versa, with respective rates. 3. Stability of the equilibrium at (0,0): The Jacobian matrix at equilibrium is the coefficient matrix $A$. Eigenvalues analysis shows positive real parts, indicating the equilibrium is an unstable focus. Analyzing System 2: $$ \frac{dx_1}{dt} = x_1, \quad \frac{dx_2}{dt} = 0.5 x_2. $$ The equilibrium is at (0,0), which is unstable as both eigenvalues are positive. The phase portrait consists of trajectories diverging away from the origin along axes. Analyzing System 3: $$ \frac{dx_1}{dt} = x_1 x_2, \quad \frac{dx_2}{dt} = x_1 + x_2. $$ Se

Mat 17 C — Luis Rademacher Midterm 2 — May 22 — 50 Minutes ✓ Solved

Mat 17 C — Luis Rademacher Midterm 2 — May 22nd — 50 minutes Open book and open notes but every student must write his or her own solution

Sample Paper For Above instruction

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Mat 17 C — Luis Rademacher Midterm 2 — May 22nd — 50 minutes Open book and open notes but every student must write his or her own solution

Sample Paper For Above instruction

References

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