Question 3Use the method of separation of variables to solve the heat equationди0 < x< 1, t>0with the following initial-boundary conditions, respectively.(a) u(t, 0) = 0, u(t, 1) = 0, u(0, x) = 2.0 < x < }0, <*<12,ди(t, 1) = 0, u(0, x) =du(b)(t,0) = 0,(c) u(t, 0) = 1, u(t, 1) = 3, u(0, x) = x? + x + 1.%3D

Answered: Image /qna-images/answer/7db1b2f6-06f8-49fe-b48e-5fdc53dc6b28.jpg

Question 3 Use the method of separation of variables to solve the heat equation du 0 < x < 1, t>0 Ət with the following initial-boundary conditions, respectively. (a) u(t, 0) = 0, u(t, 1) = 0, u(0, x) = 2. %3D du (b) (t,0) = 0, ди (t, 1) = 0, u(0, x) = 2, 0

Question 3 Use the method of separation of variables to solve the heat equation du 0 < x < 1, t>0 Ət with the following initial-boundary conditions, respectively. (a) u(t, 0) = 0, u(t, 1) = 0, u(0, x) = 2. %3D du (b) (t,0) = 0, ди (t, 1) = 0, u(0, x) = 2, 0

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Q: y"+y' - 2y = x2

A: Solution :-

Q: Ans: (x + y)e+y = C y'+--2

A: Given differential equation is homogeneous differential equation which can be solved by converting…

Q: Write the transformation matrix for a R2→ℝ2 linear transformation that rotates points 150∘ above the…

Q: Use the "mixed partials" check to see if the following differential equation is exact. (-e* sin(y) +…

A: The step by step solution of the given problem is as follows:

Q: Solve the following initial value problem for the inhomogeneous heat equation: Ut = Uxx + F(x, t), x…

Q: Ut = Uxx e-3t sin(5x) + e−²t sin x, -2t u(0,t) = u(T,t) =0, u(x,0) = sin(2x).

A: Introduction: Laplace transformation and inverse Laplace transformation is very significant in…

Q: 3. Vectors u and V are as shown, forming an angle of 30°. Describe the vector u x. = 5 units, and v…

Q: nsider the /" + 9y = 40eX; y(0) = 5, y'(0) = -2. Consider the following IVP: y"+a?y = 0; y(0) = 1,…

A: 1. Given differential equation is y'' + 9y = 40ex Auxiliary equation is given by D2 + 9 = 0 => D…

Q: Solve the IBVP for the heat equation = 2uxr, 00 e(0, 1) %3D и(4, ) 3 0, 1>0 u(x, 0) = 14sin(Tx) -…

A: Given, (1)…

Q: ди = 4 a²u 0 0, (0,t) = Ox - (1,1) = 0, t> 0, 3 u(x,0) = x, 0 <x <1.

A: Using separation of variables method to solve the heat equation.

Q: Use the method of separation of variables and the superposition principle to solve the heat equation…

Q: Solve for the formal solution [i.e., temperature distribution u(x, t)] in a metal rod of length L…

Q: The function u(x, t) = exp(-x²) is a solution of the heat equation 4rt au au² 30-3422 at ax² Use…

A: Given the solution ux,t=14πtexp-x24t of the heat equation ∂u∂t=∂2u∂x2.

Q: b) Consider the heat equation ди 2²u Ət with boundary conditions = and initial conditions ur(0, t) =…

A: The given problem is to find the heat equation partial differential equation with given initial…

Q: 5. Use the separation of variables method to solve the heat equation Ut = 0 0 u(0, t) = u(27, t)…

A: Let the solution here be u(x,t)=G(x)·Y(t), where G(x) is a function of x only and Y(t) is the…

Q: Solve the initial value problem for the heat equation with time-dependent sources au = k+ Q(x, t) at…

Q: 2. Find the solution of the one dimensional heat equation ди Ət V J²u 0 < x <L, მე2 · for the…

Q: Solve the following heat equation by means of the method of separation of variables: au ;0sxs a, tz…

A: Let us consider heat equation: ut=kuxx With conditions: u(0,t)=0, ux(l,t)=0 u(x,0)=f(x) Solution is…

Q: Find the equilibrium (SS) solutions T(x) for the heat equation T,(t,x) = kT (t,x), 0<.x< L, with…

A: Given, heat equation with boundary conditions Ttt,x=kTxxt,x 0<x<L part a boundary…

Q: The DE dy/dt = µy – oy², µ > 0 and o > 0 occurs in the study of population dynamics. It has general…

Q: Please solve the given non homogeneous heat equation ut(x,t) = uxx(x,t) − x u(0,t) = 0 u(1,t) = 0…

Q: Solve heat conduction problem in a rod : 0 <x < 2 Ut = 4uxx, Ux(0,t) = 0, Ux(2,t) = 0 %3D |3D 11n…

A: According to our guideline, we can solve only first question of multiple question. If you want…

Q: The indicated function y,(x) is a solution of the associated homogeneous equation. Use the method of…

Q: Solve the heat equation in the glass bar U; = 0,0034ux u(0, t) = 0 %3D u(24r , t) = 0 u(x, 0) = 16…

A: In this question, we use the separation of variable method. and then put x=8π

Q: Solve the following heat equation du with the boundary conditions u(0, t) = 0 u(2, t) = 0 and the…

Q: The indicated function y₁(x) is a solution of the associated homogeneous equation. y" - 5y' + 4y =…

Q: 3. Find the solution of the heat equation 2- = 0 if the boundary conditions are = 0, ôx| = 0 Ix=0…

A: The given heat equation is ∂u∂t-2∂2u∂x2=0 ,∂u∂tx=0=0,∂u∂tx=π=0 and ux,0=π2;…

Q: Find the complete solution of the DE y"- y'=4e¯* +3e* with the initial conditions that when x = 0, y…

Q: Solve the following heat equation = ut (x, t) u(0, t) = t, u(1,t) = 0, u(x,0) = x². Uxx (x, t) + x…

Q: Which is the particular solution of the differential equation y"+ p(z)y'+ q(z)y=-2z which has two…

Question

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

SEE SOLUTION Check out a sample Q&A here

Step 1

Step 2

Step 3

Step 4

Step 5

Step by step

Solved in 5 steps with 5 images

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Solving Trigonometric Equations$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.

Similar questions

The following ODE is subject to the initial conditions: y(2)=3 and y'(2)=1 d'y dy +2 --xy = 0 dx Transform the problem into a system of two 1st-order initial value ODES. • Using the Heun method and step size Ax=0.5, obtain the value of y when x = 3.0.
Solve the heat equation Ut = CUxx1 on the domaion 0 ≤ x ≤ 3, t > 0, u(x, 0) = ² - ³x², ux(0, t) = 0, X 3 2' subject to ux (3, t) = 0.
b
Solve the given initial-value problem. The DE is homogeneous. ((x^2)+(y^2))dx/dy=xy, y(-1)=2
Use the substitution u(x,t)=X(x)T(t) and separation of variables to solve the partial differential equation utt=uxNo boundary conditions are given. Take the separation constant as -a . Use the same methods used to determine the general solution to the heat equation as demonstrated in your notes and determine the most general solution in terms of a.
ди ди Exercise 1. (10 points) Consider the heat equation -(x,t) = (x, t), for x E R and > 0. at მx2 Verify if the following functions satisfy the above heat equation: u(x,t) = eax+a²t and u(x,t) = 2x³ + 6xt.
solve 3
5. Solve the heat equation ди dx? with the boundary conditions u(0, t) = 0, and u(10, t) = t and the initial value condition u(х,0) — 0.
Determine b, of the solution for the initial value problem under one-dimensional heat equation. Ut = Uxx, U (0, t) = 0 U (x, 0) = 25 00 U (5, t) = 0, t>0
2. Consider the heat equation ди k- for 0 0 with the boundary conditions u(0, t) = 0 and u(L, t) = 0, for t > 0. Solve for the following initial value conditions: a) f(x) = 4 sin ( 1 b) f(x) = x € (0, L/2] x € (L/2, L)
Please find the question attached.
Let us consider the following problem for the heat equation ди = 0 in (0,1) × (0,+∞), ốt u(x,0) = u,(x),Vx E (0,1), ди (0,t): (1) ди -(1,t) = 0, Vt > 0. Use the method of separation of variables to compute the solution of problem (1) if 1) и,(х) %3D1, Vx € (0,1). 2) u,(x)= cos77x, Vx e (0,1). 3) u,(x) = 3+2 cos7x,Vx € (0,1). 4) u,(x) = sin x, Vx e (0,1).