Solve the differential equation p 3 x – p 2 y – 1 = 0
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Sol. The given differential equation can be written as
y = px – 
... (i)
Differentiating (i) with respect to x, we get
= p + x 
+ 
⇒ p = p + x 
+ 
⇒ 
= 0
⇒ 
= 0 ... (ii)
or, x + 
= 0 ... (iii)
Now, 
= 0 ⇒ p = C
Putting p = C in (i), we obtain
y = Cx – 
This is the general solution of the given differential equation. Now, consider x + 
= 0
⇒ p 3 = – 
Eliminating p between (i) and (iii), we obtain
y = 
x– 
⇒ y = – 2 
– 
⇒ y = – 3 
⇒ y 3 = 
⇒ 4y 3 = –27x 2 This is the singular solution of the given differential equation.
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