Talk:Singular solution

To Dani, MathKnight 15:43, 5 Mar 2004 (UTC)
 * ODE = ordinary differential equation
 * PDE = partial differential equation

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 $$y = \frac{1}$$ 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 $$y=\frac{K^2}$$ 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