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**Q1:**a) Consider the Cauchy problem for the linear one-dimensional wave equation \left\{\begin{array}{ll} u_{u t}=u_{x x} & \text { for } x \in \mathbb{R} \text { and } t>0 \\ u(x, 0)=f(x) & \text { for } x \in \mathbb{R} \\ u_{t}(x, 0)=g(x) & \text { for } x \in \mathbb{R} \end{array}\right. \text { where } f \in C^{2}(\mathbb{R}) \text { and } g \in C^{1}(\mathbb{R}) \text {. Show that if } f \text { and } g \text { are odd functions, } then for every fixed t > 0, the function (0,t) is necessarily equal to 0. (i) Without proving, write down the Laplace equation in polar coordi-nates and the formula for the general solution of the Laplace equation-in a disk in R² centred at (0,0) and of radius √/6. (ii) Let D = {(x,y) € R² = r² + y² <6}. Find a harmonic function inthe disk D, satisfying u(1, y) = y + y² on the boundary of D. Write-your answer in a Cartesian coordinate system.See Answer**Q2:**Consider the Cauchy problem for the linear one-dimensional wave equation where ƒ € C²(R) and g € C¹(R). Show that if f is an odd function and g isan even function, then for every fixed t > 0, we have u₂(0, t) = f'(3t). ) Without proving, write down the Laplace equation in polar coordinates.Using the method of separation of variables, find a function u(r, 9) harmonicin the annulus (2 <r<4, 0≤ 0 ≤ 2} satisfying the boundary condition u(2, \theta)=0, \quad u(4, \theta)=\sin \theta .See Answer**Q3:**Expand the function f(x, y) = e sin(y) - 5 at (In(4), 0) by Taylor's formula up to and including the second-order terms. f(x, y) ZSee Answer**Q4:**Given f(x, y, z)= 18 x y z+18, use the total differential to estimate the change in f from the point P = (-4,-8,-4) to (-4.02, -8.02, -3.96). df =See Answer**Q5:**b. What is fyy? fyy(x, y) c. For what values of B does f satisfy the Laplace equation fax + fyy=0?See Answer**Q6:**Decide whether following functions satisfy the wave equationSee Answer**Q7:**Find the value of c such that f(x, y, t) = ect.sin(4x) .cos(2y) satisfies the heat flow equation ft = fxx + fyy. C =See Answer**Q8:**Given f(x, y, z)= 18 x y z +18, use the total differential to estimate the change in f from the point P = (-4,-8,-4) to (-4.02,-8.02, -3.96). " df =See Answer**Q9:**Decide whether following functions satisfy the wave equation utt =a² uxx.See Answer**Q10:**Find the value of c such that f(x, y, t) = e^ct.sin(4x) cos(2y) satisfies the heat flow equation ft = fxx + fyy. c=See Answer**Q11:**a. Find ztt for z(x, y, t) = cos(√9+16t) sin(3x) sin(4y). b. Does u = sin(√25 t) sin(3x) sin(4y) satisfy the membrane equation utt = uxx + uyy ?See Answer**Q12:**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=See Answer**Q13:**Consider the function f(x, y) = In (√√Ax² + 6y²). a. Find fxx. fxx (x, y) = See Answer**Q14:**or if z y x y 2 x 3 find the partial derivatives dz dx and dz dy remember not to use any decimal points in your answerSee Answer**Q15:**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).See Answer**Q16:**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) = 50See Answer**Q17:**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) = 0See Answer**Q18:**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 23See Answer**Q19:**3. Solve the following ODEs using variation of parameters. (a) 4y" - 4y + y = e/2√1-2² (b) y"-2y+y= 1+x² 2012 derSee Answer**Q20:**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. QuiSee Answer

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