Talk:Singular solution

Template:WikiProject banner shell 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
$y^{\prime }=-2y^{\frac{3}{2}}$
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 $K\to \pm \mathrm{\infty }$.

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

Pedro