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Four Bugs Differential Equation

  • #1

Homework Statement



Four bugs are walking on a flat surface. They start at the four points (1,1), (1,-1), (-1,1), and (-1,-1) and begin walking counterclockwise, each one following the next. Show that the motion of bug A, who starts at (1,1), satisfies the equation [tex]\frac{dy}{dx}[/tex]= [tex]\frac{y-x}{x-y}[/tex] and solve using an appropriate substitution.

Homework Equations



After some extensive research, I've found the same problem on another website (http://legacy.lclark.edu/~istavrov/diffeq-revsheet1-09.pdf)
that states the motion of the bug starting from (1,-1) satisfies [tex]\frac{dy}{dx}[/tex]= [tex]\frac{y-x}{x+y}[/tex] and that each bugs motion can be shown by the graph:
4bs.png
.

The Attempt at a Solution


A hint from the website, as well as from my teacher, was given saying that y=ux would be the most viable substitution. So starting there I have
[tex]y=ux => dy=xdu + udx[/tex]
[tex]\frac{xdu + udx}{dx}[/tex] = [tex]\frac{ux-x}{x-ux}[/tex]

which after some algebra leads me to:
[tex](x-xu+u)du[/tex]= [tex](u^{2}+u-1)dx[/tex]
This is where I'm stuck, and it doesn't even help me with the first part about proving WHY dy/dx is what it is (i have ideas floating around in my head but nothings clicking yet or coming together in any meaningful way). Any and all help is greatly appreciated!
 
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Answers and Replies

  • #2
Office_Shredder
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Are you sure the equation for the first part is right? Because [tex] \frac{y-x}{x-y}=-1[/tex] which doesn't make for a very interesting differential equation
 
  • #3
Are you sure the equation for the first part is right? Because [tex] \frac{y-x}{x-y}=-1[/tex] which doesn't make for a very interesting differential equation
Yes that is how he had it written, but why did you put it equal to -1?
 
  • #4
Also let it be noted that the graph I included obviously isn't correct cause it needs to be rotated 45 degrees
 

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