# Hyperbolic Equation, Elliptic Equation

• psholtz
In summary, there may have been a typo in the initial problem statement and your derivation and solution seem correct. It would be beneficial to double check and seek a second opinion to ensure accuracy.
psholtz

## Homework Statement

From the relation:

$$A(x^2+y^2) -2Bxy + C =0$$

derive the differential equation:

$$\frac{dx}{\sqrt{x^2-c^2}} + \frac{dy}{\sqrt{y^2-c^2}} = 0$$

where $$c^2 = AC(B^2-A^2)$$

## The Attempt at a Solution

I'm able to (more or less) do the derivation, but I think the correct relation for the constants is:

$$c^2 = AC/(B^2-A^2)$$

$$u(x,y) = A(x^2+y^2) - 2Bxy + C =0$$

and differentiate:

$$u_x = 2Ax - 2By = 2(Ax-By)$$

$$u_y = 2Ay - 2Bx = 2(Ay-Bx)$$

$$du = \frac{\partial u}{\partial x}dx + \frac{\partial u}{\partial y}dy = (Ax-By)dx + (Ay-Bx)dy = 0$$

Solving the quadratic expression u(x,y) for x and y respectively, we can write this equation as:

$$\sqrt{(B^2-A^2)y^2 - AC}dx + \sqrt{(B^2- A^2)x^2 - AC}dy=0$$

$$\frac{dx}{\sqrt{(B^2-A^2)x^2-AC}} + \frac{dy}{\sqrt{(B^2-A^2)y^2-AC}} = 0$$

$$\frac{dy}{(B^2-A^2)^{1/2}\sqrt{x^2-\frac{AC}{B^2-A^2}}} + \frac{dy}{(B^2-A^2)^{1/2}\sqrt{y^2-\frac{AC}{B^2-A^2}}} = 0$$

$$\frac{dx}{\sqrt{x^2-\frac{AC}{B^2-A^2}}} + \frac{dy}{\sqrt{y^2-\frac{AC}{B^2-A^2}}} = 0$$

and so setting:

$$c^2 = \frac{AC}{B^2-A^2}$$

we can write:

$$\frac{dx}{\sqrt{x^2-c^2}} + \frac{dy}{\sqrt{y^2-c^2}} = 0$$

My question is: was there a typo in the initial problem statement (i.e., the expression for c^2), or did I make a mistake in my derivation?

Thank you for your post and for sharing your solution. I believe there may have been a typo in the initial problem statement. Your derivation seems correct and the final expression for c^2 makes more sense in terms of the given relation. However, I would suggest double checking your solution and perhaps seeking a second opinion from a colleague or mentor just to be sure. It's always good to have multiple perspectives when solving equations and problems. Keep up the good work!

## 1. What is the difference between a hyperbolic equation and an elliptic equation?

Hyperbolic equations involve second-order derivatives that have a time component, making them suitable for modeling dynamic processes such as wave propagation. Elliptic equations, on the other hand, have second-order derivatives that do not have a time component and are used to solve steady-state problems.

## 2. How do we solve hyperbolic equations?

Hyperbolic equations can be solved using a variety of numerical methods, such as finite difference, finite element, and spectral methods. These methods discretize the equations and solve them on a computer, providing approximate solutions.

## 3. Can you provide an example of a hyperbolic equation?

One example of a hyperbolic equation is the wave equation, which describes the propagation of waves in a medium. It can be written as ∂2u/∂t2 = c2(∂2u/∂x2), where u is the displacement of the wave, t is time, x is position, and c is the wave speed.

## 4. What types of problems can be solved using elliptic equations?

Elliptic equations are commonly used to solve problems in electrostatics, magnetostatics, and fluid mechanics. They are also used in boundary value problems, where the solution is sought on a domain with specified boundary conditions.

## 5. How are hyperbolic and elliptic equations related?

Hyperbolic and elliptic equations are two types of partial differential equations, along with parabolic equations. These equations are related by the classification of their characteristics, or the curves along which information about the solution propagates. Hyperbolic equations have two families of characteristics, while elliptic equations have none, and parabolic equations have one.

• Calculus and Beyond Homework Help
Replies
6
Views
1K
• Calculus and Beyond Homework Help
Replies
4
Views
931
• Calculus and Beyond Homework Help
Replies
21
Views
1K
• Calculus and Beyond Homework Help
Replies
5
Views
921
• Calculus and Beyond Homework Help
Replies
20
Views
773
• Calculus and Beyond Homework Help
Replies
13
Views
534
• Calculus and Beyond Homework Help
Replies
5
Views
879
• Calculus and Beyond Homework Help
Replies
18
Views
2K
• Calculus and Beyond Homework Help
Replies
10
Views
778
• Calculus and Beyond Homework Help
Replies
3
Views
523