Understanding Exact Differential Equations: Definition and Application

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In summary, an exact differential equation is a classification of a differential equation based on its properties. It refers to the exact derivative of a function and can be represented in the form of a differential form. The term "homogeneous" has two different meanings in differential equations, depending on whether it is a first order or higher order linear equation. In general, mathematics is self-contained and does not need to refer to anything in nature.
  • #1
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I am having confusion, so from what I understand for an exact DE

dy/dx = some DE and you can manipulate it so that (y/x) appears and substitute that for v when you say y=vx, then just use separation of variables and solve.

However, what is the significance of an exact DE, why is it useful, or is it just a way that someone has figured out how to solve and bares no meaning beyond being a method to solve that type of DE?
 
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  • #2
It's a classification of a DE according to it's properties - the label is applied without reference to the method of solving it. i.e. you don't need to see if it can be solved by the method for exact DEs to discover it is exact.

It's like polynomials of order 2 are called "quadratics" - quadratics have particular characteristics and you have a standard set of tools for dealing with them.

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[edit]
The nomenclature of "exact differential equation" refers to the exact derivative of a function.
http://en.wikipedia.org/wiki/Exact_differential_equation
 
  • #3
Back in Calculus, you should have seen the concept of an "exact differential". If we have a function, f(x, y), with x and y both depending on the parameter, t, we could replace them with their expression in terms of t and find df/dt. Or, we could use the "chain rule":
[tex]\frac{df}{dt}= \frac{\partial f}{\partial x}\frac{dx}{dt}+ \frac{\partial f}{\partial y}\frac{dy}{dt}[/tex]
We can then change to "differential form"
[tex]df= \frac{df}{dt}dt= (\frac{\partial f}{\partial x}\frac{dx}{dt}+ \frac{\partial f}{\partial y}\frac{dy}{dt})dt= \frac{\partial f}{\partial x}dx+ \frac{\partial f}{\partial y}dy[/tex]
Notice that the last form,
[tex]df= \frac{\partial f}{\partial x}dx+ \frac{\partial f}{\partial y}dy[/tex]
has no mention of "t" and is true irrespective of any parameter.

Of course, if I have an equation that says "df= 0" then I can immediately write "f(x,y)= C" for some constant C.

But I can write g(x,y)dx+ h(x,y)dy that are NOT, in fact, "exact differentials".
If [itex]df= \frac{\partial f}{\partial x}dx+ \frac{\partial f}{\partial y}dy= g(x,y)dx+ h(x, y)dy[/itex], and g and h have, themselves, continuous derivatives, we must have [itex]\partial g/\partial y= \partial^2 f/\partial x\partial y= \partial h/\partial x[/itex]
If that is NOT true, gdx+ hdy is just "pretending" to be a differential- it not an "exact differential". If it is true, then gdx+ hdy= 0 is an "exact differential equation" and, once we find the correct f(x,y), we can write the general solution to the equation as f(x,y)= Constant.
 
  • #4
I feel dumb haha, I said what is an exact DE but what I described is homogeneous. I guess what is both really
 
  • #5
The phrase "homogeneous differential equation" is used in two distinctly different ways. A first order differential equation can be written in the form dy/dx= f(x, y) for some function f. The function is said to be "homogeneous" of degree n if [itex]f(\lambda x, \lambda y)= \lambda^n f(x, y)[/itex]. The differential equation dy/dx= f(x,y) is "homogeneous" if and only if f is "homogeneous".

But "homogeneous" has a different meaning for higher order linear equations. A linear differential equation can be written in the form [itex]f_n(x)d^ny/dx^n+ f_{n-1}(x)d^{n-1}y/dx^{n-1}+ \cdot\cdot\cdot+ f_1(x)dy/dx+ f_0(x)y= g(x)[/itex]. Such an equation is "homogenous[/b] if and only if the function on the right (the only one not multiply the function y or a derivative of it) is 0.

Those definitions can be found in any differential equations textbook or on Wikipedia at http://en.wikipedia.org/wiki/Homogeneous_differential_equation.
 
  • #6
I get these definitions (I've read the wikipedia articles), but I don't feel like I really understand them on anymore than a superficial level. Yeah I can spout off the definition and carry out a calculation, but do I really know the heart and soul of it??

For the exact DE, what is the potential function supposed to represent, maybe in a real life situation (mathematics rigorous definitions are incomprehensible to me).
 
  • #7
Look at the relation between gravitational field and gravitational potential.

However, mathematics is completely self-contained and need not refer to anything in nature. It is all about the relationships between numbers.
 

What are exact differential equations?

An exact differential equation is a type of differential equation where the total derivative of the equation can be expressed as the exact derivative of a function. This means that the equation can be solved by finding this function and solving for its derivative.

What is the difference between exact and inexact differential equations?

The main difference between exact and inexact differential equations is that inexact differential equations cannot be solved by finding a function whose derivative is the total derivative of the equation. Inexact differential equations require additional techniques, such as integrating factors, to solve.

What is the process for solving an exact differential equation?

The process for solving an exact differential equation involves finding a function whose derivative is equal to the total derivative of the equation. This function is known as the "integrating factor" and can be found by using a formula or by inspection. Once the integrating factor is found, the equation can be solved by integrating both sides.

What are some real-life applications of exact differential equations?

Exact differential equations have many applications in physics, engineering, and economics. They can be used to model systems that involve rates of change, such as population growth, chemical reactions, and electrical circuits. They are also useful in optimization problems, where the goal is to find the maximum or minimum of a function.

Are there any limitations to using exact differential equations?

One limitation of exact differential equations is that they can only be used to solve equations that are exact. This means that not all differential equations can be solved using this method. Additionally, exact differential equations can become more complicated when dealing with higher-order derivatives or when the function is not easily identifiable.

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