Talk:Singular solution

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To Dani,

MathKnight 15:43, 5 Mar 2004 (UTC)

[edit] Singular solution not always tangent to general solution curves

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