Solve Clairaut's Equation: Find Singular Solution

  • Context: MHB 
  • Thread starter Thread starter Suvadip
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Discussion Overview

The discussion revolves around finding the singular solution of a Clairaut's equation given by the expression $$y=px+\sqrt{1+2p^4}$$ where $$p=\frac{dy}{dx}$$. Participants explore methods for expressing the solution and the form it should take, including parametric and implicit forms.

Discussion Character

  • Technical explanation, Mathematical reasoning, Debate/contested

Main Points Raised

  • One participant requests assistance in finding the singular solution of the Clairaut's equation.
  • Another participant inquires about the progress made and provides links to external resources for further reading.
  • A participant mentions they can express the variables as functions of $$p$$ but struggles to eliminate $$p$$ to find a direct relationship between $$x$$ and $$y$$.
  • A later reply suggests that expressing the singular solution in parametric form is possible, providing specific equations for $$x$$ and $$y$$ in terms of $$p$$.
  • There is a question raised about the necessity of expressing the singular solution in implicit form versus parametric form.

Areas of Agreement / Disagreement

Participants have not reached a consensus on the best approach to express the singular solution, with differing opinions on whether implicit or parametric forms are preferable.

Contextual Notes

Some limitations include the challenge of eliminating $$p$$ to establish a direct relationship between $$x$$ and $$y$$, as well as the ambiguity regarding the preferred form of the singular solution.

Who May Find This Useful

Individuals interested in differential equations, particularly Clairaut's equations, and those looking for methods to find singular solutions may find this discussion beneficial.

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

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

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

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