Transforming a 1st Order PDE into a 2nd Order PDE: A Simple Example

In summary, we discussed finding the differential equation for the implicit equation of an ellipsoid, which can be done by taking the first and second derivatives of the equation. However, there are an infinite number of differential equations that can have the same solution as the implicit equation, and the question of whether to use a parametric or explicit version was also brought up.
  • #1
dvs77
12
0
form the differential equation for the follwing equation
(x^2)/a^2 +(y^2)/b^2+(Z^2)/c^2 =1
 
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  • #2
Just to keep us busy or what?

Show us your work!
 
  • #3
I read that as:

Form the differential equation for the following equation:

[tex] \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1 [/tex]

I don't get it...how do you get a PDE out of that?
 
  • #4
No,find the PDE whose solution is the ellipsoid's implicit equation...

Daniel.
 
  • #5
I see, ok. That wasn't really clear before.
 
  • #6
I mean u get a First order or second order partial differential equation
 
  • #7
Well for this last question : suppose you have a 1st order PDE, then just derive again and you get a 2nd order one...

The question is a bit ambigous...let's take a simpler exemple, nonparametric :

y(x)=x^2

Then there are an infinity of differential equation having that solution :

y'=2Sqrt(y)

y''=y'/Sqrt(y)

aso...

Moreover I don't know if you want the parametric equation or the explicit version...(ie. x=x(theta,phi) or x=x(y,z)...)

Just plug in this in your equation and differentiate...
 

Related to Transforming a 1st Order PDE into a 2nd Order PDE: A Simple Example

1. What is a partial differential equation (PDE)?

A PDE is a mathematical equation that involves multiple independent variables and their partial derivatives. It is used to describe relationships between these variables and their rates of change.

2. What are some applications of partial differential equations?

PDEs are used in many fields of science and engineering, such as physics, chemistry, biology, and economics. They can be used to model physical phenomena, such as heat transfer, fluid dynamics, and quantum mechanics, as well as to solve optimization problems and analyze financial markets.

3. How do you solve a partial differential equation?

The method for solving a PDE depends on its specific form and the boundary conditions. Common techniques include separation of variables, numerical methods, and the use of Green's functions. In some cases, a PDE may not have an exact solution, and numerical approximations must be used.

4. What is the difference between a partial differential equation and an ordinary differential equation?

A PDE involves multiple independent variables, while an ordinary differential equation (ODE) only involves one independent variable. Additionally, the derivatives in a PDE are partial derivatives, which consider the rate of change of one variable while holding the others constant, while derivatives in an ODE consider the rate of change of one variable with respect to itself.

5. What are some common types of partial differential equations?

Some common types of PDEs include the wave equation, heat equation, Laplace equation, and Schrödinger equation. These equations have different forms and are used to model different physical phenomena, such as wave propagation, heat transfer, and quantum mechanics.

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