Question

7.4 Solve the Laplace equation u = 0 in the square 0 < x, y < π, subject to the boundary condition u(x, 0) = u(x, π) = 1, u(0, y) = u(π, y) = 0.

Answer #1

(PDE)
Solve the ff boundary value problems using Laplace Equationnon
the square , omega= { o<x<phi, 0< y <phi}:
u(x,0) =0, u(x,phi) = 0 ; u(0,y)= siny , u(phi,y) =0

Once the temperature in an object reaches a steady state, the
heat equation becomes the Laplace equation. Use separation of
variables to derive the steady-state solution to the heat equation
on the rectangle R = [0, 1] × [0, 1] with the following Dirichlet
boundary conditions: u = 0 on the left and right sides; u = f(x) on
the bottom; u = g(x) on the top. That is, solve uxx +
uyy = 0 subject to u(0, y) =...

Solve the equation y''-y'=-3 y(0)=0 and y'(0)=0 by using laplace
transforms

Use the Laplace transform to solve the given equation.
y'' − 8y' + 20y = tet, y(0) =
0, y'(0) = 0

Solve the system of differential equations using Laplace
transform:
y'' + x + y = 0
x' + y' = 0
with initial conditions
y'(0) = 0
y(0) = 0
x(0) = 1

Solve the Homogeneous differential equation
(7 y^2 + 1 xy)dx - 1 x^2 dy = 0
(a) A one-parameter family of solution of the equation is y(x)
=
(b) The particular solution of the equation subject to the
initial condition y(1) =1/7.

Use the Laplace transform to solve the given initial-value
problem. y'' + y = f(t), y(0) = 0, y'(0) = 1, where f(t) = 0, 0 ≤ t
< π 5, π ≤ t < 2π 0, t ≥ 2π

Consider the Bernoulli equation dy/dx + y = y^2, y(0) = −1
Perform the substitution that turns this equation into a linear
equation in the unknown u(x).
Solve the equation for u(x) using the Laplace transform.
Obtain the original solution y(x). Does it sound familiar?

Solve the following differential equation by Laplace transforms.
The function is subject to the given conditions.
y''-y=5sin2t, y(0)=0, y'(0)=6

Solve the below boundary value equation
1. Ut=2uxx o<x<pi 0<t
2. u(0,t) = ux(pi,t) 0<t
3. u(x,0) = 1-2x 0<x<pi

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