Ok guys bear with me because this will be a long post.. I'm confused a little bit because every time I look at a different source material that tries to explain how to test if an equation is homogeneous, they are always explaining it in a different way, although I know that they are actually meaning the same thing, and some I can't understand while others' explanations I can, by the way i have finished integral and differential calculus class but not a straight A student so there are rough patches sometimes.. Let me show you:
First I will say the way my teacher explained it (at least how i understood it)
1. first transpose all the terms to one side so that the other side will be equal to 0.
2. Then multiply and divide both sides by the lcd so that there is no denominator
3. factor out all of the dy and all of the dx
4. If it takes the form M(x,y)dx + N(x,y)dy such that M(x,y) and N(x,y) both have the same degree, and the terms in it all have the same exponent, then the equation is homogeneous.
Is my understanding of my teacher's explanation correct? I'm also still not sure how to apply this,especially when I am running into situations where I am yielding dy^2, dx^2..
Now here is what I understood from the explanation from a book "Advanced Engineering Mathematics" by O'Neil
1. You have to equate it to y'
My problem is that sometimes there appears a (y')^2 so that means that in order to equate it to y' i have to use the quadratic formula?
2. See if it can have the form y'=f(y/x) by using algebraic manipulation
My problem is that it doesn't give enough tips and examples showing the "manipulations" that should be done..
Here is the explanation from sosmath.com
first it says that
The differential equation
is homogeneous if the function f(x,y) is homogeneous, that is-
Recognize that your equation is an homogeneous equation; that is, you need to check that f(tx,ty)= f(x,y), meaning that f(tx,ty) is independent of the variable t;
---> I don't know how to do what he just mentioned, which is how to check that f(tx,ty)= f(x,y)
---> He doesn't show the testing of the equations in his examples he just goes straight to substituting y=ux
Also how did all the different explanations above end up meaning the same thing? Or are they all three actually different ways of testing? Is there really only one way to test or different ways?
As you can see just understanding the explanations I already have difficulty.. but I have also tried to test equations taken from some books for homogeneity myself so let me show you where I am having difficulty with examples.
I have to scan what I have written on paper because I don't know how to use LaTex
In the following I am just trying to verify the homogeneity of the equations
First is three problems that I have tried to test using my teacher's method
Did I correctly place them in the form M(x,y)dy + N(x,y)dx = 0?
If so then I am correct in saying they are all not homogeneous?
Here are the same problems but this time I am trying to follow the method in the book which is to place them in the form y'=f(y/x)
Just equating it to y' I already don't get it since it's giving me (y')^2.. also I am not sure in the third problem if it already counts as the form y'=f(y/x)
Please help me fellow citizens of mathhelpforum