Does this not create solutions out of nowhere? x(dy/dx) = y^2 - y

frank789

Junior Member
Joined
Sep 16, 2017
Messages
58
Hi all,

still new to differential equations, so forgive my ignorance.

suppose we have an equation:

x(dy/dx) = y^2 - y

Very straightforward to solve.

Solved for ln|xy/(y-1)| = C

However the question asked for the solution which intersects the points (0,1) and then a second solution which intersects (0,0).

I can solve this second part by doing e^ of both sides and throwing the ol algebra hat on however I am concerned because at one point in my solution, x and y could not be 0, and then because of manipulation, they suddenly could be. This had alarm bells going off left right and center and I do not understand how they can not be a solution and then suddenly are because of manipulation.

If I did do something wrong I apologize and would love some insight in this.

thanks in advance :)

edit: and for the sake of my weigh in, I would probably answer that there is no solution at those points.
 
Last edited:
Actually, allow me to further this with asking, does exponentiating a ln function not expand the domain of the solution?
 
What were the solutions you found? Did you check whether they are actually solutions? Sometimes special cases, where parts of the work don't make sense, actually do make sense in the end.

For example, if you found y = 0 is a solution, did you observe that x(dy/dx) = 0 and y^2 - y = 0, so it satisfies the differential equation?
 
To add to what I said, did you notice that one of the first steps you took starting with x(dy/dx) = y^2 - y was to divide by x and y? So on the way to your ln, you had already reduced the possibilities, by making it impossible for x to be 0, or for y to be 0 or 1, when those are all perfectly acceptable in the original equation! So the solutions that "came out of nowhere" had first been "put into nowhere" by your own actions! That's why you have to accept them, even though at intermediate steps they appeared to be illegal.
 
To add to what I said, did you notice that one of the first steps you took starting with x(dy/dx) = y^2 - y was to divide by x and y? So on the way to your ln, you had already reduced the possibilities, by making it impossible for x to be 0, or for y to be 0 or 1, when those are all perfectly acceptable in the original equation! So the solutions that "came out of nowhere" had first been "put into nowhere" by your own actions! That's why you have to accept them, even though at intermediate steps they appeared to be illegal.

good eye thanks! That totally makes sense. In the future I will refer to the original equation and be careful about dividing by potential zeros.

moral of the story is two wrongs make a right :)
 
good eye thanks! That totally makes sense. In the future I will refer to the original equation and be careful about dividing by potential zeros.

moral of the story is two wrongs make a right :)

Two wrongs can make a right -- if you check!

But it's still possible that someone might see an alternative method that would bypass the "wrongs", or at least explain precisely why it's ultimately legal.
 
Two wrongs can make a right -- if you check!

But it's still possible that someone might see an alternative method that would bypass the "wrongs", or at least explain precisely why it's ultimately legal.

i know eventually I will come across a case where that is true and I will think of the exasperated, depressed sigh/laugh i let out when you pointed out my mistake.

thanks again.
 
Top