Determination of Singular Solutions
Let ; phai(x,y,dy/dx)=0
be a given differential equation and
let ; f(x,y,c)=0
be its general solution .
Now it is known that the envelope of any family of curves
f(x,y,c)=0 ---------- (1)
is contained in the locus obtained on eliminating , c between (1) and
[delta f(x,y,c)]/[delta c]=0 --- (2)
Let this eliminate be ,
sai (x,y)=0 ---------- (3)
As the elimination given by (3) may represent loci other than the envelope .
It is necessary to verify if any part of locus represented by (3) is or is not a solution of the given differential equation .