Solving Clairaut ODEs
The general form of Clairaut's ODE is given by:
Clairaut_ode := y(x)=x*diff(y(x),x)+g(diff(y(x),x));
Clairaut_ode ≔ y⁡x=x⁢ⅆⅆx⁢y⁡x+g⁡ⅆⅆx⁢y⁡x
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);
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.
ode ≔ y⁡x=x⁢ⅆⅆx⁢y⁡x+cos⁡ⅆⅆx⁢y⁡x
ans ≔ dsolve⁡ode
ans ≔ y⁡x=arcsin⁡x⁢x+−x2+1,y⁡x=_C1⁢x+cos⁡_C1
Note the absence of integration constant _C in the singular solution present in the above.
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