Mathematical Methods for Physics and Engineering : A Comprehensive Guide

20.8 EXERCISES

We also note that often the same general method, used in the above example

for proving the uniqueness theorem for Poisson’s equation, can be employed to

prove the uniqueness (or otherwise) of solutions to other equations and boundary

conditions.

20.8 Exercises

20.1 Determine whether the following can be written as functions ofp=x^2 +2yonly,
and hence whether they are solutions of (20.8):

(a) x^2 (x^2 −4) + 4y(x^2 −2) + 4(y^2 −1);

(b)x^4 +2x^2 y+y^2 ;

(c) [x^4 +4x^2 y+4y^2 +4]/[2x^4 +x^2 (8y+1)+8y^2 +2y].

20.2 Find partial differential equations satisfied by the following functionsu(x, y)for
all arbitrary functionsfand all arbitrary constantsaandb:

(a) u(x, y)=f(x^2 −y^2 );

(b)u(x, y)=(x−a)^2 +(y−b)^2 ;

(c) u(x, y)=ynf(y/x);

(d)u(x, y)=f(x+ay).

20.3 Solve the following partial differential equations foru(x, y) with the boundary
conditions given:

(a)x

∂u

∂x

+xy=u, u=2yon the linex=1;

(b) 1 +x

∂u

∂y

=xu, u(x,0) =x.

20.4 Find the most general solutionsu(x, y) of the following equations, consistent with
the boundary conditions stated:

(a) y

∂u

∂x

−x

∂u

∂y

=0, u(x,0)=1+sinx;

(b)i

∂u

∂x

=3

∂u

∂y

, u=(4+3i)x^2 on the linex=y;

(c) sinxsiny

∂u

∂x

+cosxcosy

∂u

∂y

=0,u=cos2yonx+y=π/2;

(d)

∂u

∂x

+2x

∂u

∂y

=0,u= 2 on the parabolay=x^2.

20.5 Find solutions of
1
x

∂u

∂x

+

1

y

∂u

∂y

=0

for which (a)u(0,y)=yand (b)u(1,1) = 1.
20.6 Find the most general solutionsu(x, y) of the following equations consistent with
the boundary conditions stated:

(a) y

∂u

∂x

−x

∂u

∂y

=3x, u=x^2 on the liney=0;