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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.
In[1]:=
Click for copyable input
Out[1]=
The general solution to Clairaut equations is simply a family of straight lines.
This plots the solution for several values of C[1].
In[2]:=
Click for copyable input
Out[2]=