4.5.11 \(x y'(x)=y(x)+x \sin \left (\frac {y(x)}{x}\right )\)

ODE
\[ x y'(x)=y(x)+x \sin \left (\frac {y(x)}{x}\right ) \] ODE Classification

[[_homogeneous, `class A`], _dAlembert]

Book solution method
Homogeneous equation

Mathematica
cpu = 0.0554972 (sec), leaf count = 19

\[\left \{\left \{y(x)\to 2 x \cot ^{-1}\left (\frac {e^{-c_1}}{x}\right )\right \}\right \}\]

Maple
cpu = 0.011 (sec), leaf count = 29

\[ \left \{ \ln \left (\csc \left ({\frac {y \relax (x ) }{x}} \right ) -\cot \left ({\frac {y \relax (x ) }{x}} \right ) \right ) -\ln \relax (x ) -{\it \_C1}=0 \right \} \] Mathematica raw input

DSolve[x*y'[x] == x*Sin[y[x]/x] + y[x],y[x],x]

Mathematica raw output

{{y[x] -> 2*x*ArcCot[1/(E^C[1]*x)]}}

Maple raw input

dsolve(x*diff(y(x),x) = y(x)+x*sin(y(x)/x), y(x),'implicit')

Maple raw output

ln(csc(y(x)/x)-cot(y(x)/x))-ln(x)-_C1 = 0