Talk:Singular solution

To Dani,

• ODE = ordinary differential equation
• PDE = partial differential equation

MathKnight 15:43, 5 Mar 2004 (UTC)

Singular solution not always tangent to general solution curves

bc The ODE
$y'=-2y^{\frac {3}{2}}$
can be solved in several ways.

When one uses
$y={\frac {1}{\left({x+C}\right)^{2}}}$
as a general solution (only the right half of every curve satisfies the diff. eq., as only the right half has a negative derivative), it is easy to see that $y=0$ is a singular solution. As this is an asymptote to the curves, it's not really a tangent.

One can also use
$y={\frac {K^{2}}{\left({Kx+1}\right)^{2}}}$
as a general solution (again, only the right half of every curve is okay). The singular solution we found above, $y=0$, is part of this family of curves. But now, the curve
$y={\frac {1}{x^{2}}}$
is a singular solution. It is not tangent to any of the general solution curves. But it is an 'asymptotic curve' for $K\to \pm \infty$.

A singular solution as a tangent is apparently not a general rule. But I'm not an expert, so maybe I missed something.

Pedro