The differential equation `y=px+f(p)`, …………..(i)
where `p=(dy)/(dx)`,is known as Clairout's equation. To solve equation i) differentiate it with respect to x, which gives either
`(dp)/(dx)=0 rArr p =c`………….(ii)
or `x+f^(i)(p)=0`…………(iii)
The number of solution of the equation `f(x)=-1` and the singular solution of the equation `y=x(dy)/(dx)+((dy)/(dx))^(2)` is

Options

A. 1
B. 2
C. 4
D. 0

Answer

B

Explanation

`y=xp+p^(2) (p=(dy)/(dx))`………..(1)
Differentiating equation (1), wr.t.x, we get ,
`p=p+x(dp)/(dx) + 2p(dp)/(dx)`
or `(dp)/(dx) (x+2p)=0`
or `(dp)/(dx) =0` or `p=-x/2`
Eliminating p from equation (1), we get
`y=-x^(2)/2 + x^(2)/4=-x^(2)/4`
Clearly, `f(x) = -x^(2)/4=-1` has two solutions .