The discussion focuses on finding the singular solution of Clairaut's equation represented as $$y=px+\sqrt{1+2p^4}$$ where $$p=\frac{dy}{dx}$$. Participants suggest expressing the solution in parametric form, specifically $$x=\dfrac{-4p^3}{\sqrt{1+2p^4}}$$ and $$y=\dfrac{1-2p^4}{\sqrt{1+2p^4}}$$. The challenge lies in eliminating the parameter $$p$$ to establish a direct relationship between $$x$$ and $$y$$. Resources such as Wolfram MathWorld and Wikipedia are recommended for further understanding of Clairaut's equation.
PREREQUISITESMathematicians, students studying differential equations, and anyone interested in advanced calculus techniques related to Clairaut's equation.
MarkFL said:What progress have you made?
Here are some links to get you started:
Clairaut's Differential Equation -- from Wolfram MathWorld
Clairaut's equation - Wikipedia, the free encyclopedia
suvadip said:I can express as x=f(p) and y=g(p). But I can't eliminate p to find a relation between x and y.