In Problems 1–10 solve Laplace’s equation (1) for a rectangular plate subject to the given boundary conditions.
2.
Trending nowThis is a popular solution!
Chapter 12 Solutions
Differential Equations with Boundary-Value Problems (MindTap Course List)
- PROBLEM (1): If =2 z-x'y and Ā = 2 xî-3 yzj+x z°k, find A.V and A xV4 at the point ( 1,-1,1). PROBLEM (2):arrow_forward- 11) Calculate the Jacobian, J, for the change of variables x = u cos(0) – v sin(0) and yu sin(0) + v cos(0).arrow_forward3. Solve the Laplace equation inside a 60° wedge of radius 1 subjected to the boundary conditions du (г.0) 3 0 r,-|=0, 3 u(1,0) = sin(20) du r,-|=0 3 u(1,0) = sin(20) (r,0)=0 +arrow_forward
- Solve Laplace's equation, = 0,0arrow_forward2. U = Uxx OLALA u(dt)=ula, t) = ( ula, 01=0 4t=(2,0) sin a t 70arrow_forwardWhich of the following is NOT a possible solution for Laplace's equation? (a) y = (AePx + Be-P*)(Ccos py + Dsin py) (b) y = (Acos px + Bsin px)(CEPY + De PY) (c) y = (Ax + B)(Cy + D) (d) y = (A P* + Be-P*)(CePy + Depy) O a O b O carrow_forward5. The function r(t) moving in 3D space. = (2 sin (¹), 3, 2 cos (t)) parametrizes (draws) the path of a particle (a) Evaluate r(t) at t = 0, 1, 2, 3, 4.arrow_forward1. Solve Laplace's equation inside a rectangle 0 0 and H > 0), with the following boundary conditions: a) u(0, y) = g(y), u(L, y) = 0, (r, 0) = 0, and u(x, H) = 0. b) u(0, y) = g(y), u(L, y) = 0, (x,0) = 0, and (x, H) = 0.arrow_forward4. Determine when the following pairs of functions are linearly independent. (a) yı(t) = erit; y(t) = er²t, r₁,72 € R (b) y(t) = cos(at); 32(t) = sin(at), a = R (c) y₁ (t) = cosh(at); y₂(t) = sinh(at), a € Rarrow_forwardProblem - 24 : Given f(x,y) = e** Y° – x*yt, show that fæy = fyx-arrow_forwardA classical equation of mathematics is Laplace's equation, which arises in both theory and applications. It governs fluid flow, electrostatic potentials, and the steady-state distribution of heat in a conducting medium. In two dimensio a²u №²u 2 Laplace's equation is Əx u(x,y) = ex cos(-6y) = + = 0. Show that the following function is harmonic; that is, it satisfies Laplace's equ dy² Find the second-order partial derivatives of u(x,y) with respect to x and y, respectively. a²u a²uarrow_forwardEx. 5.2. Find the deflection of a rectangular membrane with sides a and b with c² = 1 for the initial deflection f(x, y) = sin 3πx a sin 4πy barrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios
- Discrete Mathematics and Its Applications ( 8th I...MathISBN:9781259676512Author:Kenneth H RosenPublisher:McGraw-Hill EducationMathematics for Elementary Teachers with Activiti...MathISBN:9780134392790Author:Beckmann, SybillaPublisher:PEARSON
- Thinking Mathematically (7th Edition)MathISBN:9780134683713Author:Robert F. BlitzerPublisher:PEARSONDiscrete Mathematics With ApplicationsMathISBN:9781337694193Author:EPP, Susanna S.Publisher:Cengage Learning,Pathways To Math Literacy (looseleaf)MathISBN:9781259985607Author:David Sobecki Professor, Brian A. MercerPublisher:McGraw-Hill Education