Clairaut Equations

A Clairaut equation is a first-order equation of the form

A remarkable feature of this nonlinear equation is that its general solution has a very simple form.

This is an example of a Clairaut equation. The warning message from Solve can be ignored. It is given because DSolve first tries to find an expression for y'[x] from the given ODE.
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The general solution to Clairaut equations is simply a family of straight lines.

This plots the solution for several values of C[1].
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