The general form of Clairaut's ODE is given by:

Clairaut_ode := y(x)=x*diff(y(x),x)+g(diff(y(x),x));

$\mathrm{Clairaut\_ode}≔y\left(x\right)=x\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)+g\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)$

where g is an arbitrary function of dy/dx. See Differentialgleichungen, by E. Kamke, p. 31. This type of equation always has a linear solution:

y(x) = _C1*x + g(_C1);

$y\left(x\right)=\mathrm{\_C1}x+g\left(\mathrm{\_C1}\right)$

It is also worth mentioning that singular nonlinear solutions can be obtained by looking for a solution in parametric form. For more information, see odeadvisor/parametric.

$\mathrm{with}\left(\mathrm{DEtools}\,\mathrm{odeadvisor}\right)$

$\left[\mathrm{odeadvisor}\right]$

$\mathrm{odeadvisor}\left(\mathrm{Clairaut\_ode}\right)$

$\left[\mathrm{\_Clairaut}\right]$

$\mathrm{ode}≔y\left(x\right)=x\mathrm{diff}\left(y\left(x\right)\,x\right)+\cos \left(\mathrm{diff}\left(y\left(x\right)\,x\right)\right)$

$\mathrm{ode}≔y\left(x\right)=x\left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)+\cos \left(\frac{\ⅆ}{\ⅆx}y\left(x\right)\right)$

$\mathrm{ans}≔\mathrm{dsolve}\left(\mathrm{ode}\right)$

$\mathrm{ans}≔y\left(x\right)=\arcsin \left(x\right)x+\sqrt{-x^2+1},y\left(x\right)=\mathrm{c\_\_1}x+\cos \left(\mathrm{c\_\_1}\right)$