Solutions to Laplace's equation

In summary, the solutions to Laplace's equation of the form ax^3 + bx^2y + cxy^2 + dy^3 are given by f(x,y) = ax^3 + 3ax^y - 3axy^2 - ay^3, with the only requirement being c=-3a and b=-3d.
  • #1
Miriverite
6
0
[solved] solutions to Laplace's equation

Homework Statement


Find all solutions f(x,y) that satify Laplace's equation that are of the form:
ax^3 + bx^2y + cxy^2 + dy^3

Homework Equations


Laplace states that fxx + fyy = 0

The Attempt at a Solution


fxx = 6ax + 2by
fyy = 6dy + 2cx
so 6ax + 2by + 6dy + 2cx = 0
(3a+c)x + (3d+b)y = 0

What do I do from here?
 
Last edited:
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  • #2
Well, if f(x,y) is a solution to Laplace's equation, then it satisfies it for all x and y...what does that tell you about (3a+c) and (3d+b)?
 
  • #3
Well then:
(3a+c) = -(3d+b) = 0
a = -d
c = -b
c = -3a
b = 3a

Plugging in:
f(x,y) = ax^3 + 3ax^y - 3axy^2 - ay^3

But if you take the double partials:
fxx = 6x
fyy = 6y

and 6x-6y =/= 0 for all real numbers...

EDIT: Wow, I'm stupid. I took the double partials incorrectly so I thought I had done it wrongly.. only I hadn't. Thanks!
 
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  • #4
c=-3a and b=-3d are necessary, but I'm not sure why you think a=-b and b=-d are.

f(x,y)=ax^3-3dx^2y-3axy^2+dy^3 satisfies Laplace's equation for all a and d not just a=d
 
  • #5
If a=-b adn b=-d, then my above function satisfies for all a - that's good enough, yes?
 
  • #6
Miriverite said:
If a=-b adn b=-d, then my above function satisfies for all a - that's good enough, yes?

It does, but its not the most general solution...which is what you are looking for.

(3a+c) = -(3d+b) = 0 does not mean that a=-b and b=-d...the only requirement is that c=-3a and b=-3d.
 
  • #7
I see. Thanks a bunch!
 

What is Laplace's equation?

Laplace's equation is a partial differential equation that describes the relationship between the second derivatives of a function in three-dimensional space.

What are solutions to Laplace's equation used for?

Solutions to Laplace's equation are used in various fields of science and engineering to model and analyze phenomena such as heat flow, fluid dynamics, and electrostatics.

How do we solve Laplace's equation?

There are several methods for solving Laplace's equation, including separation of variables, the method of images, and the use of Green's functions. The appropriate method depends on the boundary conditions and the specific problem being solved.

What are the boundary conditions for Laplace's equation?

Boundary conditions for Laplace's equation specify the values or behavior of the function at the edges or boundaries of the domain. They are essential for obtaining a unique solution to the equation.

What are some real-world applications of solutions to Laplace's equation?

Solutions to Laplace's equation have numerous applications in fields such as physics, engineering, and mathematics. They can be used to model and solve problems related to heat transfer, fluid flow, electrical potential, and many other phenomena.

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