(a) y" + 2y + y sin +3 cos2x
(b) y)-g=e²
y(0)=y'(0)=y" (0) = g(0) = 0
Fig: 1
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
Question 1 This question concerns the volume of an object bounded by a parabola x = y² - 5y on the left and a straight line x = y on the right (z and y in metres). The cross section is show below: Ty x = y? — 5y The height of the object, h(x, y) (in metres), is modelled in this assessment using the following function: h(x,y) = (y − z)(z − y? +5y). x=y (a) (5 marks) Find the maximum height by doing the following: i) find all the critical points of the function h(x, y), ii) identify the single critical point that is within the cross section area A depicted above (not including points on the boundary). iii) show that this critical point is a local maximum using the Hessian determinant test, iv) evaluate the value of h at this maximum (b) (5 marks) The volume of the object V (metres³) is defined as the integral =ff h(x, y) da V = where A is the cross-section depicted in the figure above. Calculate the volume by doing the following:/n(c) (2 marks) use MATLAB to create a contour plot of the height h(x, y). Make sure to include a few (three or more) contours (level curves) between zero and the maximum value you found in Qla.
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. 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)
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
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=
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.
1. Solve the following ODES or initial value problems using the method of undetermined coefficients. (a) y" + 2y + y sin +3 cos2x (b) y)-g=e² y(0)=y'(0)=y" (0) = g(0) = 0
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).
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