How Is a General Solution for Laplace's Equation in 2D Obtained?

[Reshma]

The Laplace's equations in 2-dimensions if V is the electric potential is given by:
$$\frac{\partial^2 V}{\partial x^2} + \frac{\partial^2 V}{\partial y^2} = 0$$
Since this is a second order partial differential equation, the simple rules of an ordinary differential do not apply. The solution will not contain a definite number of arbitrary constants. So how is a general solution obtained for this equation?

 

[hypermorphism]

See harmonic functions.

 

[starfield]

see Introduction to Electrodynamics by David J Griiffiths...
there is a very good introiduction to Partial differential equations in it...

 

[ZapperZ]

Or see Mary Boas "Mathematical Methods in the Physical Science". Chances are, if you're having problems with this, you may need to look at a bunch of other mathematical techniques in 2nd order partial differential equation and how they are used in physics. This book covers such a thing.

Zz.

 

[dextercioby]

The Laplace's equations in 2-dimensions if V is the electric potential is given by:
$$\frac{\partial^2 V}{\partial x^2} + \frac{\partial^2 V}{\partial y^2} = 0$$
Since this is a second order partial differential equation, the simple rules of an ordinary differential do not apply. The solution will not contain a definite number of arbitrary constants. So how is a general solution obtained for this equation?

1.Bring it to canonical form.
2.Identify the type of problem you're dealing with depending on the initial/boundary conditions.
3.Using the separation of variables is the easiest way to get a particular solution.

Daniel.