Question

Find singular solution of equation xp^2-(x-a)^2

Answer 1

Answer:

we see that the substitution v=x2+y2 leads to v′=2x+2py, which is a term in the ODE, so

v−(x+y)v′1+p+v′24(1+p)2=0

Taking

P=dvdu=dvdxdxdu=2(x+yp)1+p

and then

dudx=1+p⟹u=x+y

and

4v=4Pu−P2.

That's why the author says that only experience can help determining the correct change of variables. If you have terms px+y, choose v=xy and see if a proper u can be constructed. If you have terms x+py, choose v=x2+y2 and try to construct the u. Further than that, there is no other intuitive explanation that I can give, or accept.

Answer 2

Answer:

singular solution x = 0

General solution 9(y + c)^2 = 4x(x - 3a)^2