# Differential equations - how to form them?

• Dec 25th 2008, 08:24 AM
wirefree
Differential equations - how to form them?
Forum,

I seek to gain an understanding of how differential equations are arrived at.

In most cases, intuition, basic knowledge of physics, or computer simulation can tell us that the unknown variable is dependent on one or more variables. How this translates into an equation is not clear to me.

I ask because I have solved hundreds of ODE/PDEs, but given a sample real-world situation, how would I even formulate the problem, before solving it, seems to be the challenge.

Best,
wirefree
• Dec 25th 2008, 12:26 PM
mr fantastic
Quote:

Originally Posted by wirefree
Forum,

I seek to gain an understanding of how differential equations are arrived at.

In most cases, intuition, basic knowledge of physics, or computer simulation can tell us that the unknown variable is dependent on one or more variables. How this translates into an equation is not clear to me.

I ask because I have solved hundreds of ODE/PDEs, but given a sample real-world situation, how would I even formulate the problem, before solving it, seems to be the challenge.

Best,
wirefree

Read a book on the subject eg. Amazon.com: Modelling with Differential and Difference Equations (Australian Mathematical Society Lecture Series): Glenn Fulford, Peter Forrester, Arthur Jones: Books
• Dec 26th 2008, 06:53 PM
wirefree
Quote:

Originally Posted by mr fantastic

Appreciate the response, mr fantastic.

I would be much obliged should you be able to provide a little foretaste of the subject. The concept accompanied by a small example would prove beneficial.

A knack for compression is the mark of brilliance.

Best regards,
wirefree
• Dec 26th 2008, 07:13 PM
Mush
Quote:

Originally Posted by wirefree
Appreciate the response, mr fantastic.

I would be much obliged should you be able to provide a little foretaste of the subject. The concept accompanied by a small example would prove beneficial.

A knack for compression is the mark of brilliance.

Best regards,
wirefree

A small example from fluid mechanics:

An irrotational flow is defined a flow having zero vorticity. Vorticity is defined as the curl of the velocity vector ($\displaystyle \vec{V} = (V_x,V_y,V_z)$) of a flow:

$\displaystyle \nabla \times \vec{V} = 0$

An identity from vector calculus gives:

$\displaystyle curl(grad \phi) = \nabla \times \nabla \phi = 0$

Hence, for an irrotational flow, there exists a scalar function $\displaystyle \phi$ whose gradient is equal to the velocity vector of the flow.

$\displaystyle grad \phi = \vec{V}$.

Or:

$\displaystyle (\frac{d \phi}{dx},\frac{d \phi}{dy},\frac{d \phi}{dz}) = (V_x, V_y, V_z)$.

The continuty (mass conservation) equation of an incompressible flow can be written:

$\displaystyle \nabla . \vec{V} = 0$

And hence:

$\displaystyle \nabla . (\nabla \phi) = 0$

$\displaystyle => \nabla ^2 \phi = 0$

$\displaystyle => \frac{d^2 \phi}{dx^2} + \frac{d^2 \phi}{dy^2}+\frac{d^2 \phi}{dz^2} =0$.

This is Laplace's equation, which is a linear 2nd order homogeneous differential equation. It governs ALL fluid flows which are both irrotational and incompressible.

The concept: We started with the definitions of an irrotational flow, and an incompressible flow and expressed them in terms of vector calculus using the nabla vector (which is defined as: $\displaystyle \nabla = (\frac{d}{dx},\frac{d}{dy},\frac{d}{dz})$). Combining these definitions with the identity $\displaystyle curl(grad f) = 0$, we managed to deduce a linear 2nd order homogeneous differential equation which governs the physics behind ALL flows of this type.
• Jan 17th 2009, 12:07 PM
wirefree
Appreciate the response, Mush. The example you illustrated was a very sophisticated one and very well formulated at that by you. (Bow)

As closure, I will be persistent in reducing the thread down to the simplest of ODEs I fished in my course text, right off page 1, and request some thoughts on the nature & complexity of problem it might address and how it might have been formulated:

$\displaystyle dy/dx + 5y = e^x$