# Four Bugs Differential Equation

## 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 $$\frac{dy}{dx}$$= $$\frac{y-x}{x-y}$$ 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 $$\frac{dy}{dx}$$= $$\frac{y-x}{x+y}$$ and that each bugs motion can be shown by the graph:
.

## 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
$$y=ux => dy=xdu + udx$$
$$\frac{xdu + udx}{dx}$$ = $$\frac{ux-x}{x-ux}$$

which after some algebra leads me to:
$$(x-xu+u)du$$= $$(u^{2}+u-1)dx$$
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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Office_Shredder
Staff Emeritus
Are you sure the equation for the first part is right? Because $$\frac{y-x}{x-y}=-1$$ which doesn't make for a very interesting differential equation
Are you sure the equation for the first part is right? Because $$\frac{y-x}{x-y}=-1$$ which doesn't make for a very interesting differential equation