Does the homogeneous equation Ac = 0 where A =TA, have a non-trivial solution? Yes No...
3. Consider the non homogeneous heat equation ut- urr+ 1 with non homogeneous boundary conditions u(0. t) 1, u(1t) (a) Find the equilibrium solution ueqx) to the non homogeneous equation. (b) The solution w(r, t) to the homogenized PDE wt-Wra, with w(0,t,t)0 1S -1 Verify that ugen(x, t)Ue(x) +w(x, t) solves the full PDE and BCs (c) Let u(x,0)- f(x) - 2 - ^2 be the initial condition. Find the particular solution by specifying all Fourier coefficients 3. Consider the...
point) As an illustration of the difficulties that may arise in using the method of undetermined coefficients, consider 0 a. Form the complementary solution to the homogeneous equation. c(t)-c +, 02 ai -e®a, where a b. Show that seeking a particular solution of the form Walt s a constant vector does not work. In fact, if had this form, we would arrive at the following contradiction: d1 and a1- c. Show that seeking a particular solution of the form jp(t)...
6. (12 points) The simplest (homogeneous, isotropic) 'Ohmic' materials are characterized by J=oE, where o is the constant) electrical conductivity (do not confuse it with the surface charge density!), and E is the electric field within the conductor. (a) (4 points Using the time-dependent form of the continuity equation, along with Ohm's law from above, derive a differential equation that describes the time and spa- tial dependence of the volume charge density par, t) within some conductor. Charac- terize (mathematically)...
Previous Problem Problem List Next Problem (1 point) Let's find the general solution to z2y"-5zy, + 8y-(2-P) using reduction o of order (1) First find a non-trivial solution to the complementary equation z' smaller power m. 5zy' +8y0 of the form z. There are two possibilities, pe (2) Now set u = tizm and determine a first order equation (in standard form) that ,' t' must satisfy (3) Solve this for z using cl as the arbitrary constant 4) Solve...
= Consider the equation ax² + 2xy + cy? 1 where a > 0 and ac – 62 > 0. Note that a 6 Y = 1. Consider an invertible linear map y X = 6 с х 1 e и between (x, y) and (u, v) given by = y 0 1 V (a) Choose a value for e (in terms of a, b and c), so that the given equation on the (u, v) LEO plane becomes [u...
Mark each statement True or False. Justify each answer. a. A homogeneous system of equations can be inconsistent. Choose the correct answer below. O A. True. A homogeneous equation can be written in the form Ax o, where A is an mxn matrix and 0 is the zero vector in R". Such a system Ax -0 always has at least one solution, namely x-0. Thus, a homogeneous system of O B. True. A homogeneous equation cannot be written in the...
(1 point In general for a non-homogeneous problem y' + p(x) +(z) = f() assume that y. is a fundamental set of solutions for the homogeneous problemy" p(x) + (2) 0. Then the formula for the particular solution using the method of variation of parameters is where (z)/ and ()/() where W() is the Wronskian given by the determinant W (2) (2) W2) 31(2)/(2) dr. NOTE When evaluating these indefinite integrals we take the W(2) So we have the de...
Exercise 15.3 Under what conditions does the matrix equation Ax = 0 have a solution x that is NOT the zero vector where A is assumed to be square? a) never b) always c) if A is full rank d) if A is rank deficient
please explain the steps as well! it’s imp for me to understand this question. i have attached the table for last part of the question Consider the second order non-homogeneous constant coefficient linear ordinary differ- ential equation for y(x) ору , dy where Q(x) is a given function of r For each of the following choices of Q(x) write down the simplest choice for the particular solution yp(x) of the ODE. Your guess for yp(x) will involve some free parameters...
please solve all 3 Differential Equation problems 3.8.7 Question Help Consider the following eigenvalue problem for which all of its eigenvalues are nonnegative y',thy-0; y(0)-0, y(1) + y'(1)-0 (a) Show that λ =0 is not an eigenvalue (b) Show that the eigenfunctions are the functions {sin α11,o, where αη įs the nth positive root of the equation tan z -z (c) Draw a sketch indicating the roots as the points of intersection of the curves y tan z and y...