Question

Singular solution for the Clairaut’s equation \(y = y’x+\frac{a}{y’}\) is given by _______

(a) \(\frac{x^2}{a^2} + \frac{y^2}{a^2} = 1\)

(b) y^2=-4ax

(c) y^2=4ax

(d) x^2=-2ay

I had been asked this question in final exam.

I would like to ask this question from Clairaut’s and Lagrange Equations in portion Ordinary Differential Equations – First Order & First Degree of Engineering Mathematics

Answer 1

Right choice is (c) y^2=4ax

Explanation: Let \( p = y’ = \frac{dy}{dx}\)

Given equation is of the form y = px + f(p), whose general solution is y = cx + f(c)

thus the general solution is \(y = cx + \frac{a}{c}\) …………..(1), to find the value of c

we differentiate (1) partially w.r.t  ‘c’ i.e \( 0 = x-\frac{a}{c^2} \rightarrow c^2 = \frac{a}{x} \rightarrow c = \sqrt{\frac{a}{x}}\)

hence (1) becomes \(y = \sqrt{\frac{a}{x}} x + a \sqrt{\frac{x}{a}} \rightarrow y=2\sqrt{ax}\)

y^2=4ax is the singular solution.