Partial Differential Equations : Separation of Variables (6 Problems)

Please see the attached file for the fully formatted problems.

11. Separation of Variables. By usingu(x, t) = X(x)T(t) or u(x,y, t) = X(x)Y(y)T(t), separate the following PDEs into two or three ODEs for X and T or X, Y, and T. The parameters c and k are constants. You do not need to solve the equations.
NOTE: one of the equations cannot be separated. Indicate this when you discover that equation.
(a) utt = (xux)x
(b) Utt = C2Uxx
(c) Ut = k(u + Uyy)
(d) ut=k(yu+uy)
(e) Ut + cu ku
(f) Ut = k(yu + XU)

Separation of variables is demonstrated. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.

Find the region in the xy plane in which the equation [(x - y)^2 - 1] u_xx + 2u_xy + [(x - y)^2 - 1] u_yy = 0Â is hyperbolic. The complete problem is in the attached file.

1. Please see the attached file for the fully formatted problems.
a) Use separation of variables to solve
b) Solve the following exact differential equation
c) By means of substitution y=vx solve the differential equation
d) By means of the substitution for approipriate values of and , solve the d

Please help with the following problem. Provide step by step calculations for each problem.
Consider the Helmholtz partialdifferential equation:
u subscript (xx) + u subscript (yy) +(k^2)(u) =0
Where u(x,y) is a function of two variables, and k is a positive constant.
a) By putting u(x,y)=f(x)g(y), derive ordinary diff

Using the chain rule of partial differentiation, to show that the differential equation Â Â Â Â Â Â Â Â Â Â Â Â Â
Â Â Â Â Â Â Â Au_xx + Bu_xy + Â Cu_yy + Du_x + Eu_y + Fu = G Â Â Â Â Â Â Â
Â Â transforms into the differential equation Â Â Â Â Â Â Â Â Â Â Â Â
Â Â Â Â Â

What technique would be used to solve the differential equation:
You do not have to solve the differentialequations, just write what technique that you would use to solve them.
Solution techniques: Equilibrium solutions, Separation of variables, exact equations, integrating factors, substitutions for homogeous and Bernoulli

Hi,
Please help working on
section 1.1 problems 2,4,8,14,16
section 1.2 problems 6,10,20,24,27
thank you
See attached
Classify each as an ordinary differential equation (ODE) or a partialdifferential equation (PDE), give the order, and indicate the independent and dependent variables. If the equation is an ODE, ind

Please solve the following initial value problem:
dx/dt = x^2 - 4, x(0) = 0.
and find the maximal interval of existence of the solution.
We solve the initial value problem using separation of variables. We use partial fractions to solve for the integral of 1/(x^2-4).

1. Find solutions to the given Cauchy- Euler equation
(a) xy'+ y =0 (b) x2y'' + xy'+y =0 ; y(1) =1, y'(1) =0
2. Find a solution to the initial value problem
x2y' + 2xy = 0; y (1) = 2
3. Find the general solution to the given problems
(a) Y' + (cot x)y = 2cosx (b) (x-5)(xy'+3y) = 2
4. Solve t

In solving this problem, derive the general solution of the given equation by using an appropriate change of variables.
1. âˆ‚u/âˆ‚t - 2 âˆ‚u/âˆ‚x = 2
Answer: u(x,t) = f(x + 2t) - x
In this exercise, (a) solve the given equation by the method of characteristic curves, and (b) check you answer by plugging it back int