MHB Solve Clairaut's Equation: Find Singular Solution

[Suvadip]

Please help me to find the singular solution of the following Clairaut's equation

$$y=px+\sqrt{1+2p^4}$$ where $$p=\frac{dy}{dx}$$

 

[MarkFL]

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]

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.

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

 

[Fernando Revilla]

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.

Is it necessary to express the singular solution in implicit form?. You can write it in parametric form: $$x=\dfrac{-4p^3}{\sqrt{1+2p^4}},\quad y=\dfrac{1-2p^4}{\sqrt{1+2p^4}}$$