1 First order ode

Given first order ode \(F\left ( x,y,y^{\prime }\right ) =0\) the goal is find its singular solutions (if any). This method applies to first order ode’s of degree not one.

Singular solution here, is meant to be the solution that can not be obtained from the general solution (hence called \(y_{c}\left ( x\right ) \)) for any value of \(c\) (including \(\pm \infty \)). This singular solution (called \(y_{s}\left ( x\right ) \)) will be the envelope of the family of solutions of the general solution \(y_{c}\left ( x\right ) \). It will have no constant in it, unlike the general solution.

If the ode is an initial value problem, and if the uniqueness theorem says there is a unique in an interval around \(\left ( x_{0},y_{0}\right ) \) then no singular solution exists as this will violate the uniqueness theorem. The main steps used to find singular solution are the following

1. Find \(y_{s}\) using p-discriminant method by eliminating \(y^{\prime }\) from \(F\left ( x,y,y^{\prime }\right ) =0\) and \(\frac {\partial F}{\partial y^{\prime }}=0\).
2. Verify that each \(y_{s}\) found satisfies the ode.
3. Find general solution to the ode \(y_{c}\left ( x\right ) \). Written as \(\Psi \left ( x,y,c\right ) =0\)
4. Verify that the two equations \(y_{c}\left ( x_{0}\right ) =y_{s}\left ( x_{0}\right ) \) and \(y_{c}^{\prime }\left ( x_{0}\right ) =y_{s}^{\prime }\left ( x_{0}\right ) \) are satisfied at an arbitrary point \(x_{0}\) for each singular solution found in step 1. If so, then \(y_{s}\left ( x\right ) \) is singular solution. (envelope of the family of curves of the general solution).
5. An alternative to (4) which seems to be more common, is to use the c-discriminant method method. In this we work directly with the implicit general solution \(\Psi \left ( x,y,c\right ) =0\). Then eliminate \(c\) from this and the equation \(\frac {\partial \Psi \left ( x,y,c\right ) }{\partial c}=0\). Then compare the resulting \(y_{s}\) with the one found from step (1) which is the p-discriminant method. If singular solution from p-discriminant and c-discriminant is the same, then this is indeed a singular solution. If they are different, then it is not a singular solution. Only the common singular solutions from the p-discriminant and the c-discriminant are valid. If p-discriminant does not yield solution, then we will use the solution from only c-discriminant. The Examples below show how to use these methods. In all the following examples, the plots will show the singular solution(s) as thick red dashed lines.

Given ode \(F\left ( x,y,p\right ) =0\) then necessary and sufficient conditions that singular solution exist are (see E.L.Ince page 88)

1. \(F=0\)
2. \(\frac {\partial F}{\partial p}=0\)
3. \(\frac {\partial F}{\partial x}+p\frac {\partial F}{\partial y}=0\)

The above should be satisfied simultaneously. However, I am not able to verify these now. However, Ince says that \(\frac {\partial F}{\partial y}\neq 0\) is necessary for singular solution to exist. So will add this check below.