# Cartesian Coordinates Problem

Required to solve the attached Laplace Equation problems using separation of variables.

Let €:= {(x,y) : 0 < x < 1, 0 < y < 1}.

(1) Let u(x,y) satisfy the PDE

uxx + uyy = 0,

u(0, y) = 0, u(1, y) = 0, 0 < y < 1;

u(x, 0) = 0, u(x,1) = f(x), 0 < x <1.

(2) If f(x) = x, 0 < x < ½; and f(x) = ½, ½ < x <1; then what does u(x,y) from problem (1) look like?

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The differential equation is:

(1.1)

With the boundary conditions:

(1.2)

And:

(1.3)

We start by writing the function as a product of two independent functions:

(1.4)

Then, the partial differentials turn into full differentials:

(1.5)

(1.6)

The boundary conditions now become:

(1.7)

Plugging (1.5) and (1.6) back into (1.1) we get:

(1.8)

Equation (1.9) is separated. Each side of the equation is completely independent of the other side. Since it must be true for any , the two sides must be equal the same constant:

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#### Solution Summary

Cartesian Coordinates are exemplified using separation of variables.