Differential Equations - not Linear, Separable, or Exact

[inveniam_viam]

Problem and Equation: Solve dy/dx=-y/(x^2+y)

Put into standard form, this is ydx+(x^2+y)dy=0

The only ways of solving differential equations that I currently know are when they are either linear (which this is not), separable (this is also not), or exact (ditto), and I vaguely know about substitution but a little guidance on how to go about solving a problem like this would be greatly appreciated! I tried substituting V for the denominator (v=x^2+y) to try to get rid of the y but had no luck. Thanks very much!

 

[jackmell]

Any chance you made a mistake with that equation? The reason I ask is I tried this and that for a little while then in desperation, turned to Mathematica's DSolve and it returns a very complicated inverse function for it's solution.

 

[inveniam_viam]

I asked the professor about it and apparently it's just an equation he "thought of off the top of his head" to see if we could solve it, and then told us to do it for homework, lol. Do you have any clue how to go about getting to that "very complicated inverse function" for the solution?

 

[jackmell]

Ok, I meant "it's in implicit form" and not really "inverse". Find a machine running Mathematica, better yet, spend this entire semester learning how to use Mathematica if you don't already, say one hour a week at least. Then type:

In[274]:=
DSolve[Derivative[1][y][x] == 
   -y[x]/(x^2 + y[x]), y, x]

And it returns:

During evaluation of In[274]:= Solve::tdep:The equations appear to involve the variables to be solved for in an essentially non-algebraic way. >>

During evaluation of In[274]:= Solve::tdep:The equations appear to involve the variables to be solved for in an essentially non-algebraic way. >>

Out[274]=
Solve[C[1] == 
   -((-2*x*BesselJ[0, 
        I*Sqrt[4*x^2 - 4*(x^2 + y[x])]] - 
      I*BesselJ[1, I*Sqrt[4*x^2 - 
           4*(x^2 + y[x])]]*
       Sqrt[4*x^2 - 4*(x^2 + y[x])])/
     (-2*x*BesselY[0, (-I)*
         Sqrt[4*x^2 - 4*(x^2 + y[x])]] + 
      I*BesselY[1, (-I)*Sqrt[4*x^2 - 
           4*(x^2 + y[x])]]*
       Sqrt[4*x^2 - 4*(x^2 + y[x])])), 
  y[x]]

That's essentially telling you that the solution it found is in implicit form, kinda like for example:

y^3+6x^2+2-2y=0

and it's asking you to use "Solve" to solve for y[x].

But the implicit solution it found is too complicated to do symbolically else Mathematica would have done that. So take that expression and just say let that C[1] constant equal to 1 for starters. Then plot that implicit function using ContourPlot. That gives you a set of points which satisfy the relation. Now take those points and arrange them in the form of (x,y) pairs, then do a Fit on that data to arrive at an approximate representation of the solution.

See what I mean, spend time learning Mathematica. It will help you a great deal in your math studies.