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 = 0.0059319 (sec), leaf count = 17
\[\left \{\left \{y(x)\to \text {InverseFunction}[f,2,2]\left [c_1,0\right ]+c_1 x\right \}\right \}\]
Maple ✓
cpu = 0.158 (sec), leaf count = 63
\[ \left \{ y \left ( x \right ) ={\it \_C1}\,x+{\it RootOf} \left ( f \left ( {\it \_C1},{\it \_Z} \right ) \right ) ,[x \left ( {\it \_T} \right ) ={\frac {D_{{1}} \left ( f \right ) \left ( {\it \_T},{\it RootOf} \left ( f \left ( {\it \_T},{\it \_Z} \right ) \right ) \right ) }{D_{{2}} \left ( f \right ) \left ( {\it \_T},{\it RootOf} \left ( f \left ( {\it \_T},{\it \_Z} \right ) \right ) \right ) }},y \left ( {\it \_T} \right ) ={\frac {D_{{1}} \left ( f \right ) \left ( {\it \_T},{\it RootOf} \left ( f \left ( {\it \_T},{\it \_Z} \right ) \right ) \right ) {\it \_T}+D_{{2}} \left ( f \right ) \left ( {\it \_T},{\it RootOf} \left ( f \left ( {\it \_T},{\it \_Z} \right ) \right ) \right ) {\it RootOf} \left ( f \left ( {\it \_T},{\it \_Z} \right ) \right ) }{D_{{2}} \left ( f \right ) \left ( {\it \_T},{\it RootOf} \left ( f \left ( {\it \_T},{\it \_Z} \right ) \right ) \right ) }}] \right \} \] 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),'implicit')
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))