Slope Field

[goku900]

If you plug in the slope field dy/dx = x^2/y^2 into this generator, http://www.math.rutgers.edu/~sontag/JODE/JOdeApplet.html


How do you interpret that graph? Is that straight line that runs through the origin part of the solution?

 

[HallsofIvy]

There are an infinite number of "solutions" to that equation. The "straight line that runs through the origin" is one of those solutions. It is, in fact, the graph of the equation y= x. And it is easy to see that is a solution: if y= x, then dy/dx= 1 while x/y= 1 so that (x/y)^2= 1.

It is relatively easy to solve that differential equation: if dy/dx= y^2/x^2, then y^{-2}dy= x^{-2}dx and, integrating, -y^{-1}= -x^{-1}+ C so that y= 1/(1/x+ C)= x/(Cx+ 1). Different values of x give different solutions (C= 0 gives y= x) each graph of which is tangent to those "slope field" lines at each point on the graph.