Use the method of Laplace transforms and a translation of y(t) to solve the IVP. y^{\prime \prime}+2 y^{\prime}-3 y=4 e^{t}+5 \sin t, y\left(\frac{\pi}{2}\right)=0, y^{\prime}\left(\frac{\pi}{2}\right)=1

2. A support for electrified railway cables is cantilevered from the side of the track by a beam with spring stiffness k. The mass of the beam is 3M and is assumed to be concentrated at the free end. The cable, of mass 2M, is supported by a spring of stiffness k from the end of the cantilever. The system of equations governing the motion of the system is: 3 M y_{1}^{\prime \prime}=-2 k y_{1}+k y_{2} 2 M \ddot{y}_{2}=k y_{1}-k y_{2} k = 22 Write the above system of differential equations in matrix form. Then, by considering the trial solution: y = e"X, show that system can be written as an eigenvalue problem. (3) b) Find the general solution for the system of equations by solving the eigenvalue problem. (12)

3. Green's functions (Haberman § 9.3, see problems 9.3.9 and 9.3.11) Consider d'u dr²+u = f(x) subject to subject to u(0) = 0, u(x/2) = 0. (14) The goal in (a) is to find an integral representation for the unknown u(r) of the form u(x) = ™² G(E,x)ƒ (E)d£ (15) where G(r, ) is the Green's function. Note that (15) only holds for homogeneous boundary conditions (e.g. (14)). (a) Solve for G(§, z) directly from JG (§, x) მ2 (13) +G(§, x) = 8(§ - x) (16) G(0,r)=0 G(T/2, x) = 0. (17) You will need to determine and apply the matching conditions at = r as discussed in lecture to find G(z, E) (see also Haberman page 388).

3. The vibration of a cable supporting a suspension bridge can be described by the one- dimensional wave equation, \frac{\partial^{2} u}{\partial t^{2}}=\alpha^{2} \frac{\partial^{2} u}{\partial x^{2}} The problem has the following boundary and initial conditions: • Is the trial solution u(x,t) = g(t)sin(-x) sensible for this problem, discuss10why/why not. (3) \text { Using the trial solution } u(x, t)=g(t) \sin \left(\frac{n \pi}{10} x\right), \text { convert the wave equation } into a single ODE and find its general solution. (6) c) Write the general solution to the PDE and solve for the unknown constants. (6)

A violin string produces a vibration where we have that A > 0, that a and b are arbitrary constants, and that utt = c²uxx for some c. Find E>0 in terms of A. E=

2. Consider the following differential equations. Determine the form of the particular solution, g,. for use in the method of undeter- mined coefficients. Simply find the form of the particular solution without solving for the coefficients. Remember to check for duplication with solutions to the homogeneous equation. (a) 4y"+y=t-008 () (b) "5y+6y=cost-te (c) "" "t²te^ (d) y(4)ytet + sint 23

1. For a cylindrical coordinate, write the partial differential equation for the followings: a. the Heat Equation b. the Steady State Equation, c. the Wave Equation. 2. Determine solution of the following partial differential equation 8²U/ax² = a² du/dt = 0 subjected to: U (x,0) = 0 U (o, t) = 50 U (1,t) = 50

Consider the following ODE with given IC: Y^{\prime}(x)=x^{2} \cos (Y(x))^{2}, Y(0)=1 \text { and answer the following questions: } \text { What is } \frac{\partial f(x, z)}{\partial z} ? In what region of x will the solution exist? c) Find the analytical solution Y(x) and verify where it exists.

4. Consider the nonconservative mass-spring system governed by +2 +26x = 0, z(0) = 1, ż(0) = 4 (a) Find the solution z(t) and its derivative (t), and evaluate z(7/5) and a(w/5). (b) Calculate the total energy E(t) of the system when t = x/5. (c) Calculate the energy loss in the system due to friction in the time interval from t = 0 tot = x/5. Qui

1 Using equation (5.3b), find the Taylor series around x = -pi/2 of the function f(x) = sin(2x), up to and including the term in h². Remember not to use any decimal points in your answer. f(xo + h) ~