Differential function vs differential equation

In summary, the conversation discusses the difference between differentiation and differential equations. Differentiation involves finding the rate of change at a specific point, while differential equations involve finding a function that satisfies an equation with derivatives. Differential equations can be more complex, especially when dealing with non-linear equations, and may require the use of a computer. Additionally, the conversation clarifies the meaning of a differential function and explains that it is typically presented implicitly in differential equations.
  • #1
Square1
143
1
As the title suggests, could someone "differentiate" :) between the two phrases?

We learn about differentiation first and find out that you can get the rate of change at a certain point 'a'. Then we find out that you can obtain a function that pumps out the derivative at a defined point. What is the the difference between the differential function and, the latest topic we've started, differential equations?

Thanks
 
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  • #2
Square1 said:
As the title suggests, could someone "differentiate" :) between the two phrases?

We learn about differentiation first and find out that you can get the rate of change at a certain point 'a'. Then we find out that you can obtain a function that pumps out the derivative at a defined point. What is the the difference between the differential function and, the latest topic we've started, differential equations?

Thanks

Hey Square1.

The big difference IMO is that one is explicit and another is implicit.

Consider the function f(x) = y = x^2 + x + 2 and the function y^2x + x^2SQRT(y+x) = y. The first is an explicit equation of y in terms of x and the second is an implicit equation of y and x in terms of each other.

In a differential equation, you usually get an implicit equation in the general case in terms of the derivatives by remembering that d/dx of dy/dx is d^2y/dx^2 and so on. It is this idea that will help you understand the extension of dy/dx = f(x) to something like d^2y/dx^2 - yxdy/dx + x^2 + 2y = 0.

If you ever do this in depth, you'll see that this implicit nature makes things really complicated especially if you are not dealing what is known as a linear differential equation.

Because real world systems are modeled often with non-linear equations, we need to know how to solve them and often this means using a computer. But the thing is we can't just plug everything into a computer: we need to know theoretically what the computer needs to calculate in order for the output on our computer to even make sense and we also need to know when the DE we have even makes sense to begin with: in other words, the DE itself might not make sense as a unique function and if this is the case then we can't even compute the function because it's not really a sound function to begin with.
 
  • #3
Hey thanks for the reply. It took me some to let the stuff sink in/do some more problems to increase my "feel" for the matter. So your response has more meaning now :)
 
  • #4
I can't tell you any difference because I don't think I have ever seen the phrase "differential function"! Do you mean the differential of a function or differentiable function?

Of course, the crucial part of any kind of equation is the fact that there is a "=" in it! A differential equation is an equation that includes the derivative of a function. Typically, but not always, the "problem" associated with a differential equation is to find the function whose derivative satisfies that equation.
 
  • #5
yeah I think it's nothing more complicated than that (the definition that is!). And as chiro pointed out, they often in my classes are presented implicitly. If they were in explicit form, it would be really no additional work then just doing basic integration that we've done up till now I think.
 

1. What is the difference between a differential function and a differential equation?

A differential function is a mathematical function that relates the output value to the input value, whereas a differential equation is a mathematical equation that relates the rate of change of a variable to its current value. In simpler terms, a differential function describes a relationship between two variables, while a differential equation describes how one variable changes in relation to another.

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

Differential functions and equations are used in various scientific fields such as physics, engineering, and biology to model and predict the behavior of systems. They are particularly useful in studying systems that involve changing variables, such as population growth, chemical reactions, and motion.

3. Can you give an example of a differential function and a differential equation?

An example of a differential function is the position function, which relates an object's position to time. An example of a differential equation is the Newton's second law of motion, which relates an object's acceleration to the forces acting on it.

4. How do you solve a differential equation?

The method for solving a differential equation depends on the type of equation and the variables involved. Some common methods include separation of variables, substitution, and integrating factors. In some cases, it may be necessary to use numerical methods to approximate a solution.

5. What are the applications of solving differential equations?

Solving differential equations allows us to make predictions and understand the behavior of complex systems in various fields of science and engineering. It is also used in designing and optimizing systems, such as in creating models for climate change, chemical reactions, and electrical circuits.

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