Explore BrainMass
Share

# Partial Differential Equations : Separation of Variables (6 Problems)

This content was STOLEN from BrainMass.com - View the original, and get the already-completed solution here!

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)

https://brainmass.com/math/partial-differential-equations/partial-differential-equations-separation-of-variables-6-problems-129527

#### Solution Summary

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.

\$2.19

## First order differential Equations, partial DE's

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 the Bernoulli equation y' = xy3 - 4y

5. Use separation of variables to solve the verhulst population problem

N' (t) = (a-bN) N, N (0) = N0; a,b > 0

6. Verify that each of the given functions is a solution of the given differential equation, and then use the Wronskian to determine linear dependence/ independence
Y''' - y''- 2y' = 0 {1, e-x, e2x}

View Full Posting Details