How are differential equations developed

In summary, the order of a differential equation is determined by what the derivative is representing. For "dynamics" problems, which involve motion, the equation is typically second order, while problems involving "growth" or "rate of flow" are usually first order. "Elasticity" problems tend to be fourth order due to physical reasons.
  • #1
JaredPM
20
0
When differential equations are being developed, what dictates the order of the differential? What decides if it is second order, third order, fourth order, fifth order, etc...
I understand that the process is taking the derivative of a derivative of a derivative, but what decides if the the third, fourth, or fifth derivative has any value when performing the analysis of process being investigated?
 
Physics news on Phys.org
  • #2
It depends upon what the derivative is supposed to represent. "Dynamics" problems, about motion, depend upon "F= ma" and acceleration, a, is the second derivative of the position function so if the problem is to determine position from a given force function, the differential equations are typically second order. On the other hand, problems involving just "growth" or "rate of flow", because those are just "rate of change", tend to be first order differential equations.

Problems involving "elasticity", on the other hand, but again from physical reasons tend to be fourth order equations.
 

1. What is the history behind the development of differential equations?

Differential equations have been around for centuries, with the earliest known use dating back to the ancient Greek mathematician Pythagoras in the 6th century BC. However, it wasn't until the 17th and 18th centuries that mathematicians such as Isaac Newton and Gottfried Leibniz began to formally develop the concepts and techniques of differential equations as we know them today.

2. How are differential equations used in science and engineering?

Differential equations are used in a wide range of scientific and engineering fields, including physics, chemistry, biology, economics, and engineering. They are used to model and understand complex systems and processes, such as the motion of objects, the behavior of chemical reactions, and the spread of diseases. They also play a critical role in the development of technologies such as airplanes, satellites, and computer simulations.

3. What are the basic types of differential equations?

There are two main types of differential equations: ordinary differential equations (ODEs) and partial differential equations (PDEs). ODEs involve a single independent variable and are used to describe one-dimensional systems, while PDEs involve multiple independent variables and are used to describe multidimensional systems. Other types of differential equations include linear and nonlinear, as well as first-order and higher-order equations.

4. What are the key techniques used to solve differential equations?

There are several techniques used to solve differential equations, including separation of variables, substitution, integration, and series expansion. The specific technique used depends on the type of differential equation and its complexity. In many cases, numerical methods are also used to approximate solutions to difficult or impossible-to-solve equations.

5. How are differential equations related to calculus?

Differential equations and calculus are closely related, as differential equations are essentially equations that involve derivatives. In fact, solving a differential equation often involves using calculus techniques, such as integration and differentiation. Understanding calculus is essential for understanding and working with differential equations.

Similar threads

A Solution to a differential equation with variable coefficients
    • Differential Equations
Replies
6
Views
1K
I Finding Max and Min Extremes of a Function with Second Derivatives Equal to Zero
    • Differential Equations
Replies
1
Views
1K
I Feynmann's connection machine model
    • Differential Equations
Replies
1
Views
692
B Basic Idea of Differential Equations
    • Differential Equations
Replies
25
Views
2K
I How do I solve this first order second degree differential equation?
    • Differential Equations
Replies
7
Views
3K
I Second order non-homogeneous linear ordinary differential equation
    • Differential Equations
Replies
2
Views
2K
I On vibrating string and differentiating infinite sum
    • Differential Equations
Replies
7
Views
380
A Integrability of Systems of Differential Equations
    • Differential Equations
Replies
5
Views
986
I Fundamental matrix of a second order 2x2 system of ODEs
    • Differential Equations
Replies
2
Views
1K
B A few differential equation questions
    • Differential Equations
Replies
19
Views
2K

Hot Threads

Recent Insights

Back
Top