Partial differential equation Homework Help

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) Z

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 =

b. What is fyy? fyy(x, y) c. For what values of B does f satisfy the Laplace equation fax + fyy=0?

Decide whether following functions satisfy the wave equation

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 =

Decide whether following functions satisfy the wave equation utt =a² uxx.

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=

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 ?

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=

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 answer

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).

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

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

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

5. Convert the following initial value problems into a system of first order initial value problems. Write your answer in the form u' Au + f, u(0) = uo. (a) (b) +ty"+y=1, y(0)=0, g'(0)=1, /'(0) = 2 y+3y + 2z=e, y(0)=0, (0)=1 z"+y+22=1, 2(0)=1, '(0)=0 Copy

2) Verify that the indicated function is an explicit solution of the given differential equation. Assume an appropriate interval I of definition for each solution. y"-6y + 13y=0; y = ³ cos 2x

6) Solve the given differential equation by using an appropriate substitution. d x + 3y 3x + y

2. Solve the IVP y" - y" -y'+y=6et, y(0) = y'(0) = y" (0) = 0 using Laplace Transforms.