p(x)y^n + q(x)y^w = c

[Vol]

I am really confused. What does this mean: p(x)y^n + q(x)y^w = c ?

this is a DE. Thus, n and w are order of derivative. 1st derivative, 2nd derivative, etc.

I can see you are supposed to solve for y. What I don't get is isn't y = f(x)? And p(x) is a function too. So, looks like you have a relation between the functions. p(x), y^n, q(x), and y^(w). And you have to solve for y. What does this mean? Can somebody explain it in plain English? p(x) and q(x) are coefficients? Why would you mate them with y? Because you are saying function x function + function x function = constant. Why do this? Are there any practical examples?

 

[Dr.Peterson]

I am really confused. What does this mean: p(x)y^n + q(x)y^w = c ?
I can see you are supposed to solve for y. What I don't get is isn't y = f(x)? And p(x) is a function too. So, looks like you have a relation between the functions. p(x), y^n, q(x), and y^(w). And you have to solve for y. What does this mean? Can somebody explain it in plain English? p(x) and q(x) are coefficients? Why would you mate them with y? Because you are saying function x function + function x function = constant. Why do this? Are there any practical examples?

Well, no, you can't just see that you are supposed to solve for y; All you have shown is an equation, which could be used in many different ways. Someone has to tell you the goal, and probably some other facts. Please quote the entire problem, including any instructions that were given.

We can guess that y is intended to be a function of x, that p and q are given functions, and that n, w, and c are constants; but we don't know that. And in general, there is no algebraic way to solve for y in terms of all these things.

So you'll have to tell us the context. Evidently you found this in connection with something about differential equations, judging from your choice of a forum. Tell us more!

 

[mmm4444bot]

… isn't y = f(x)?

We don't know, yet, but it's certainly possible that y is also a function of x.

… p(x) and q(x) are coefficients?

Yes, definitely. Each of them represents a factor (number) multiplying a derivative of y.

Why would you mate them with y? Because you are saying function x function + function x function = constant. Why do this? Are there any practical examples

Yes, there are many practical examples all around us (eg: mathematics, engineering, science, technology, economics).

Here's a contrived example:

The height formula (as a function of elapsed time) for a projectile near Earth's surface (ignoring stuff like air friction, spin, aerodynamics) is:

h(t) = 1/2∙g∙t^2 + v₀∙t + h₀

where g is gravitional constant for Earth, v₀ is initial velocity vector (launch angle) and h₀ is initial height.

If NASA needed to determine the height of a projectile on some other planet, they would need to change g. In fact, they might be interested in several planets, so g is a function of p (a set of planets).

h(t) = 1/2∙g(p)∙t^2 + v₀∙t + h₀

Furthermore, the initial velocity vector might be a function of temperature, which is a function of time. In such a case, v₀ depends on time. The initial height might be a function of different machines (x) firing the projectile.

h(t) = 1/2∙g(p)∙t^2 + v(t)∙t + h(x)

Modeling stuff in the real world gets complicated, and we use experiences with simpler situations to work our way through more difficult ones. :cool:

 

[Vol]

Well, this is a DE. Thus, n and w are order of derivative. 1st derivative, 2nd derivative, etc.

 

[Dr.Peterson]

Well, this is a DE. Thus, n and w are order of derivative. 1st derivative, 2nd derivative, etc.

What you wrote was not a differential equation; it is, now that you have explained your nonstandard notation. That is an important part of asking a question.

But what do you want to know? The meaning or application of a particular equation comes from its context, and you haven't yet told us where you found this and what was said about it. What I can say is that it is a rather general form for a differential equation, of which something like the following would be an example: (x^2 - 1)y" + (3x+2)y' = 5.

Whether there are any specific applications in which such an equation arises is separate from the question of how to solve it; but I would imagine if you were told about it, it would be motivated either by the fact that some equation of this form has been needed some time in history, or just by curiosity! Again, what was said?

 

[HallsofIvy]

Well, this is a DE. Thus, n and w are order of derivative. 1st derivative, 2nd derivative, etc.

"mth" and "nth" Derivatives are commonly denoted \(\displaystyle y^{(m)}\) and \(\displaystyle y^{(n)}\).