Sol. The given differential equation is x = yp + ap 2 ... (i)

Differentiating both sides of (i) with respect to y, we get

= p + y + 2ap

⇒ = p + (y + 2ap)

⇒ –p = (y + 2ap)

⇒ = +

⇒ + y = ... (ii)

This is a linear differential equation in p and y having

I.F. = dp = =

Multiplying both sides of (ii) by I.F. and integrating with respect to, we get

y = dp

⇒ y = –2a dp

⇒ y = –2a dp

⇒ y = –2a dp

⇒ y = – a [p + log

(p + )] – 2a log (p + ) + C

⇒ y = – ap – 3a log

(p + ) + C ... (iii)

Eliminating p between (i) and (iii), we obtain the required solution.