282 Chapter 4 The Potential Equation
homogeneous or “homogeneous-like” conditionsonoppositesidesofagener-
alized rectangle. Examples of “homogeneous-like” conditions are the require-
ment that a function remain bounded as some variable tends to infinity, or the
periodic conditions atθ=±π(see Section 4.5). The point is that if two or
more functions satisfy the conditions, so does a sum of those functions.
In spite of the limitations of the method of separation of variables, it works
well on many important problems in two or more variables and provides in-
sight into the nature of their solutions. Moreover, it is known that in those
cases where separation of variables can be carried out, it will find a solution if
one exists.
EXERCISES
1.Classify the following equations.a.∂^2 u
∂x∂y=0;b. ∂(^2) u
∂x^2
+ ∂
(^2) u
∂x∂y
+∂
(^2) u
∂y^2
= 2 x;
c. ∂
(^2) u
∂x^2 −
∂^2 u
∂x∂y+
∂^2 u
∂y^2 =^2 u;
d.
∂^2 u
∂x^2 −^2
∂^2 u
∂x∂y+
∂^2 u
∂y^2 =
∂u
∂y;
e. ∂
(^2) u
∂x^2
−∂
(^2) u
∂y^2
−∂u
∂y
=0.
2.Showthat,inpolarcoordinates,anannulus,asector,andasectorofan
annulus are all generalized rectangles.
3.InwhichoftheequationsinExercise1canthevariablesbeseparated?
4.Sketch the regions listed in the text as generalized rectangles.
5.Solve these three problems and compare the solutions.a. ∂(^2) u
∂x^2
+∂
(^2) u
∂y^2
=0, 0 <x<1, 0 <y,
u(x, 0 )=f(x),0<x<1,
u( 0 ,y)=0, u( 1 ,y)=0, 0 <y;
b.
∂^2 u
∂x^2 =
∂^2 u
∂y^2 ,0<x<1,^0 <y,