Two func- Problem 1. Calculate the third mixed partial fay for the following tions, by choosing t...
1. Consider the Partial Differential Equation ot u(0,t) = u(r, t) = 0 a(x, 0)-x (Y), sin (! We know the general solution to the Basic Heat Equation is u(z,t)-Σ b e ). n= 1 (b) Find the unique solution that satisfies the given initial condition ur, 0) -2. (Hint: bn is given by the Fourier Coefficients-f(z),sin(Y- UsefulFormulas/Facts for PDEs/Fourier Series 1)2 (TiT) » x sin aL(1)1 a24(부) (TiT) 1)+1 0
1. Consider the Partial Differential Equation ot u(0,t) =...
Consider the partial differential equation together with the boundary conditions u(0, t) 0 and u(1,t)0 for t20 and the initial condition u(z,0) = z(1-2) for 0 < x < 1. (a) If n is a positive integer, show that the function , sin(x), satisfies the given partial differential equation and boundary conditions. (b) The general solution of the partial differential equation that satisfies the boundary conditions is Write down (but do not evaluate) an integral that can be used to...
Consider a 2 m long metal rod. The temperature u(z,t) at a point along the rod at any time t is found by solving the heat equation k where k is the material property. The left end of the rod ( 0) is maintained at 20°C and the right end is suddenly dipped into snow (0°C). The initial temperature distribution in the rod is given by u(x,0)- (i) Use the substitution u(z,t) ta,t)+20-10z to reduce the above problem to a...
1. Consider the insulated heat equation up = cum, 0 <r<L, t > 0 u (0,t) = u (L, t) = 0, t > 0 u(x,0) = f(2). What is the steady-state solution? 2. Solve the two-dimensional wave equation (with c=1/) on the unit square (i.e., [0, 1] x [0,1) with homogeneous Dirichlet boundary conditions and initial conditions: (2, y,0) = sin(x) sin(y) (,y,0) = sin(x). 3. Solve the following PDE: Uzr + Uyy = 0, 0<<1,0 <y < 2...
Problem 1. Find the general solution of an ID heat equation: Tt(x,t) = 4Txx(x,t) with the boundary conditions T(0,t) = T(2,t) = 0. Note that T(x,t) denotes the temperature profile along x of a uniform rod of length 2. Problem 2. Solve the following ID wave equation: Ott(x,t) = 0xx(x,t) with the boundary conditions 0 (0,t) = 0;(1,t) = 0, where 0(x,t) refers to the twist angle of a uniform rod of unit length. Problem 3. Show that the solution...
Whats the answer to number
1?
1. Let r(t) = -i-e2t j + (t? + 2t)k be the position of a particle moving in space. a. Find the particle's velocity, speed and direction at t = 0. Write the velocity as a product of speed and direction at this time. b. Find the parametric equation of the line tangent to the path of the particle at t = 0. 2. Find the integrals: a. S (tezi - 3sin(2t)j + ick)...
what is the answer for number
4
1. Let r(t) = -i-e2t j + (t? + 2t)k be the position of a particle moving in space. a. Find the particle's velocity, speed and direction at t = 0. Write the velocity as a product of speed and direction at this time. b. Find the parametric equation of the line tangent to the path of the particle at t = 0. 2. Find the integrals: a. S (tezi - 3sin(2t)j +...
Question 1 [Total 20 marks] (a) [5 marks] In a steady-state two-dimensional heat flow problem, the temperature, u, at any point in the domain (t, ) satisfies the differential equation u y(2-y) u= U0F With the given temperature boundary condition as follows: u(x, 0) = 0, u(x, 2) = x(4-x), 0 < x < 4 Calculate the temperature at the interior points a, b, and c using a mesh size h-1.
Question 1 [Total 20 marks] (a) [5 marks] In...
Problem 5. (1 point) A Bernoulli differential equation is one of the form +P()y= Q()y" (*) Observe that, if n = 0 or 1, the Bernoulli equation is linear. For other values of n, the substitution u =y- transforms the Bemoulli equation into the linear equation + (1 - x)P(3)u = (1 - x)^(x). Consider the initial value problem ry' +y = -3.xy?, y(1) = 2. (a) This differential equation can be written in the form (*) with P(1) =...
webwork math233 17742 huang s19 16.4 parametrized surfaces /1 16.4 Parametrized surfaces: Problem 1 Problem List Next Problem Previóus Problem (1 point) Show that Ф (u, u)-- (9u t-3, u-u, 17u t u) parametrizes the plane 2x-y-z--6. Then (a) Calculate Tu. T. and n(u, v) (b) Find the area of S-D(D), where D = (mu) : 0 < u < 6,0 < u < 7 (c) Express f(x, y, 2 yz in terms of u and v and evaluate JTs...