Question

(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 it. Same for 5(b) below.

Answer #1

(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.

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)

Use the method for solving Bernoulli equations to solve the
following differential equation
dx/dy+5t^7x^9+x/t=0
in the form F(t,x)=c

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

Suppose u(t,x) and v(t,x ) is C^2 functions defined on R^2 that
satisfy the first-order system of PDE Ut=Vx,
Vt=Ux,
A.) Show that both U and V are classical solutions to the wave
equations Utt= Uxx.
Which result from multivariable calculus do you need to justify
the conclusion.
B)Given a classical sol. u(t,x) to the wave equation, can you
construct a function v(t,x) such that u(t,x), v(t,x)
form of sol. to the first order system.

Solve heat equation for the following conditions
ut = kuxx t > 0, 0 < x < ∞
u|t=0 = g(x)
ux|x=0 = h(t)
2. g(x) = 1 if x < 1 and 0 if x ≥ 1
h(t) = 0;
for k = 1/2

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.

Solve the following partial differential equation:
yux + xuy = 0
u(0,y) = e- y^2

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

(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

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