Math Is Fun Forum

I am not a master of the theory behind calculus. Are differential operators a thing that can be canceled out the same way a variable may be canceled?

If we want to set the derivative of f equal to the reciprocal of the derivative of the inverse of f, we might write

where f-inverse is a function x in terms of y. That is,

Now I know that in differential equations one can move the dx over to the other side the equation when separating the variables. Does that mean that other operations can be used as well? Such as cancellation?

If we cross-multiple we can get

and if d is a thing that may be canceled out, we could then get

and, if so, since

we can reduce to

That may be a complete perversion of mathematics. Sorry, if so.

Does what I just did make sense? Is that one way of showing how the function y can be taken from the two derivatives without calculus?

Thanks so much for the help.

Here is a problem from a handout I tried.

I got

Is anyone getting similar answers?

P. S. Thanks for the help.

Here is my attempt at exercise 1 on page 3 of that pdf you sent. How does this look?

Which yields that λ is 1 ± i.

So that

And by the same logic it can be found that

If I am doing this correctly.

How does that look?

The idea is to compute eigenvectors by substituting the eigenvalues in for λ one at a time. Then you enter the two into a general solution of the form

where L1 and L2 are the eigenvalues since the forum won't let me use λ in math text. V1 and V2 are the eigenvectors.

The initial problem, however, was not in finding a general solution but in recognizing how eigenvalues affect direction fields. It seems like such a nebulous business.

Some of these problems take forever and the concept of eigenvalues seems really vague. I was hoping someone had a list of clear-cut, straight-forward rules for how eigenvalues affect a system. I do not like it when math is sketchy.

Right now we are dealing with linear solutions. Supposing you have a linear system in terms of dy/dt and dx/dt you are asked to judge, just by the eigenvalues, how the system would act if plotted as a direction field.

For example, say you have the system

and you compute the eigenvalue to be -4, if I am even doing that part correctly (I don't know how to show my work with matricies on this forum) then you find the system, if plotted in Maple, rotates clockwise as a wheel would, rather than spiraling toward or away from the origin.

Here is an image from Maple to demonstrate my point (attached).

I am required to be able to tell, just from calculating the eigenvalues (and without Maple) what the system will do. Will it rotate? Will it go toward the center? away from the origin? Both?

Thanks. And I am good, thank you. How are you?

Edit: This won't let me upload my picture. Or I am unable to see my uploaded picture.

I was going to try to find position functions for both of them but Leonardo is hard to do because of limiting information. I keep ending up with too many unknowns.

That could potentially pose as a problem.

I see that.

Might all of this be best put into terms of position rather than velocity? Could that in some way be made to work with the positions of the cars?

In your's, where did

come from?

I don't understand the answer either. Or how you arrived at it.

I am working on it, too, but with none much luck thus far.

So I understand, you are saying that the fact I got the same answer as you is a coincidence?

I'll work it out your way and see what I get.

Hi gAr,

I believe so. I have this:

If I am understanding your question.