In each of Problems 1 through 6, determine whether the method

of separation of variables can be used to replace the given partial

differential equation by a pair of ordinary differential equations. If so,

find the equations.

1. xuxx + ₁ = 0

2 tuxx+xu=0

3. Uxx+Uxt + U₁ = 0

4. (p(x)ux)x-r(x)U₁ = 0

5.

6.

Uxx + (x+y)uyy = 0

Uxx + Uyy + xU = 0

7. Find the solution of the heat conduction problem

100μxx = Ut

u(0, 1) = 0,

u(x, 0) = sin(2x) - sin(5mx), 0≤x≤1.

8. Find the solution of the heat conduction problem

Uxx = 4u₁, 00;

u(0, 1) = 0, u(2, 1) = 0, 1 > 0;

u(x,0) = 2 sin(xx/2) = sin(x) + 4 sin(2x), 0≤x≤2.

Consider the conduction of heat in a rod 40 cm in length whose ends

are maintained at 0°C for all > 0. In each of Problems 9 through 12,

find an expression for the temperature u(x, t) if the initial temperature

distribution in the rod is the given function. Suppose that a ² = 1.

9. u(x,0) = 50, 0

J.x,

0 ≤ x < 20,

20 ≤ x ≤ 40

10. u(x,0) =

0 < x < 1, t> 0;

u(1, 1) = 0, t> 0;

40-x,