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) Which of the following is true about solutions of differential equation `y=xy^(')+sqrt(1+y^('2))`?

Options

A. the general solution of equation is family of parabolas
B. the general solution of equation is family of circles
C. the singular solution of equation is circle
D. the singular solution of equation is ellipse

Answer

C

Explanation

Given equation can be rewritten as
`y=xp+sqrt((1+p^(2))), p=(dy)/(dx)`……….(1)
Differentiating w.r.t x, we get
`p=p+x(p)/(dt) + 1/(2sqrt(1+p^(2))2p (dp)/(dx)`
`i.e,(dp)/(dx)=0` or `p/sqrt(1+p^(2))=-x`
i.e., `p=c` or `p=x/sqrt(1-x^(2))`
Thus, `y=cx+ sqrt((1+c^(2))` gives the general solution and `x^(2)+y^(2)=1` as singular solution.