ODE
\[ f\left (y'(x),y(x)-x y'(x)\right )=0 \] ODE Classification
[_Clairaut]
Book solution method
Clairaut’s equation and related types, \(f(y-x y', y')=0\)
Mathematica ✓
cpu = 0.169076 (sec), leaf count = 17
\[\{\{y(x)\to \text {InverseFunction}[f,2,2][c_1,0]+c_1 x\}\}\]
Maple ✓
cpu = 0.541 (sec), leaf count = 63
\[\left [\left [x \left (\textit {\_T} \right ) = \frac {D_{1}\left (f \right )\left (\textit {\_T} , \RootOf \left (f \left (\textit {\_T} , \textit {\_Z}\right )\right )\right )}{D_{2}\left (f \right )\left (\textit {\_T} , \RootOf \left (f \left (\textit {\_T} , \textit {\_Z}\right )\right )\right )}, y \left (\textit {\_T} \right ) = \frac {D_{1}\left (f \right )\left (\textit {\_T} , \RootOf \left (f \left (\textit {\_T} , \textit {\_Z}\right )\right )\right ) \textit {\_T} +D_{2}\left (f \right )\left (\textit {\_T} , \RootOf \left (f \left (\textit {\_T} , \textit {\_Z}\right )\right )\right ) \RootOf \left (f \left (\textit {\_T} , \textit {\_Z}\right )\right )}{D_{2}\left (f \right )\left (\textit {\_T} , \RootOf \left (f \left (\textit {\_T} , \textit {\_Z}\right )\right )\right )}\right ], y \left (x \right ) = \textit {\_C1} x +\RootOf \left (f \left (\textit {\_C1} , \textit {\_Z}\right )\right )\right ]\] Mathematica raw input
DSolve[f[y'[x], y[x] - x*y'[x]] == 0,y[x],x]
Mathematica raw output
{{y[x] -> x*C[1] + InverseFunction[f, 2, 2][C[1], 0]}}
Maple raw input
dsolve(f(diff(y(x),x),y(x)-x*diff(y(x),x)) = 0, y(x))
Maple raw output
[[x(_T) = D[1](f)(_T,RootOf(f(_T,_Z)))/D[2](f)(_T,RootOf(f(_T,_Z))), y(_T) = (D[1](f)(_T,RootOf(f(_T,_Z)))*_T+D[2](f)(_T,RootOf(f(_T,_Z)))*RootOf(f(_T,_Z)))/D[2](f)(_T,RootOf(f(_T,_Z)))], y(x) = _C1*x+RootOf(f(_C1,_Z))]