Question

(a) Separate the following partial differential equation into two ordinary differential equations: Utt + 4Utx − 2U = 0. (b) Given the boundary values U(0,t) = 0 and Ux (L,t) = 0, L > 0, for all t, write an eigenvalue problem in terms of X(x) that the equation in (a) must satisfy.

Answer #1

(a) Separate the following partial differential equation into
two ordinary differential equations: e 5t t 6 Uxx + 7t 2 Uxt − 6t 2
Ut = 0. (b) Given the boundary values Ux (0,t) = 0 and U(2π,t) = 0,
for all t, write an eigenvalue problem in terms of X(x) that the
equation in (a) must satisfy. That is, state (ONLY) the resulting
eigenvalue problem that you would need to solve next. You do not
need to actually solve...

The heat equation is: ∂u/∂t = α∇^2u
where α is the thermal diffusivity of a substance, and u(x, y,
z, t) describes the temperature of a substance as a function of
space and time. Show that the separation of variables procedure can
be successfully applied to this partial differential equation to
separate the space and time variables. You should obtain, as your
final answer, a series of ordinary differential equations.

Series Solutions of Ordinary Differential Equations For the
following problems solve the given differential equation by means
of a power series about the given point x0. Find the recurrence
relation; also find the first four terms in each of two linearly
independed sollutions (unless the series terminates sooner). If
possible, find the general term in each solution.
y"+k2x2y=0, x0=0,
k-constant

Partial differential equations
Solve using the method of characteristics
ut +1/2 ux + 3/2 vx = 0 , u(x,0) =cos(2x)
vt + 3/2 ux + 1/2 vx = 0 , v(x,0) = sin(2x)

(1 point) Given the following differential equation
(x2+2y2)dxdy=1xy,
(a) The coefficient functions are M(x,y)= and N(x,y)= (Please input
values for both boxes.)
(b) The separable equation, using a substitution of y=ux, is
dx+ du=0 (Separate the variables with x with dx only and u with du
only.) (Please input values for both boxes.)
(c) The solution, given that y(1)=3, is
x=
Note: You can earn partial credit on this
problem.
I just need part C. thank you

Solve the following wave equation using Fourier Series
a2uxx = utt, 0 < x < L, t
> 0, u(0,t) = 0 = u(L,t), u(x,0) = x(L - x)2,
ut(x,0) = 0

(PDE)
WRITE down the solutions to the ff initial boundary problem for
wave equation in the form of Fourier series :
1. Utt = Uxx ; u( t,0) = u(t,phi) = 0 ; u(0,x)=1 , Ut( (0,x) =
0
2. Utt = 4Uxx ; u( t,0) = u(t,1) = 0 ; u(0,x)=x , Ut( (0,x) =
-x

Walter A . Strauss- Partial Differential Equations (2nd
Edition)
Chapter 7.1, Problem 10E
Let u(x，y) be the harmonic function in the unit disk with
boundary
values u(x， y) = x^2 on {x^2 + y^2 = 1}. Find the Rayleigh-Ritz
approximation of the
form, w0 + c1w1 = x^2 + c1(1 - x^2 - y^2).

Write the second order differential equation as a system of two
linear differential equations then solve it.
x''-6x'+13x=0 x(0)= -1 x'(0)=1

Use C++ in Solving Ordinary Differential Equations using
a
Fourth-Order Runge-Kutta of Your Own Creation
Assignment:
Design and construct a computer program in C++ that will
illustrate the use of a fourth-order
explicit Runge-Kutta method of your own design. In other words, you
will first have to solve the Runge-Kutta equations of condition for
the coefficients
of a fourth-order Runge-Kutta method. See the
Mathematica notebook on solving the equations for 4th order RK
method.
That notebook can be found at...

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 19 seconds ago

asked 2 minutes ago

asked 24 minutes ago

asked 44 minutes ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 2 hours ago

asked 2 hours ago

asked 2 hours ago

asked 2 hours ago

asked 2 hours ago