ODE
\[ x y'(x)=y(x)-2 x \tanh \left (\frac {y(x)}{x}\right ) \] ODE Classification
[[_homogeneous, `class A`], _dAlembert]
Book solution method
Homogeneous equation
Mathematica ✓
cpu = 0.3599 (sec), leaf count = 16
\[\left \{\left \{y(x)\to x \sinh ^{-1}\left (\frac {e^{c_1}}{x^2}\right )\right \}\right \}\]
Maple ✓
cpu = 0.35 (sec), leaf count = 34
\[\left [y \left (x \right ) = \arctanh \left (\frac {1}{\sqrt {-x^{4} \textit {\_C1} +1}}\right ) x, y \left (x \right ) = -\arctanh \left (\frac {1}{\sqrt {-x^{4} \textit {\_C1} +1}}\right ) x\right ]\] Mathematica raw input
DSolve[x*y'[x] == -2*x*Tanh[y[x]/x] + y[x],y[x],x]
Mathematica raw output
{{y[x] -> x*ArcSinh[E^C[1]/x^2]}}
Maple raw input
dsolve(x*diff(y(x),x) = y(x)-2*x*tanh(y(x)/x), y(x))
Maple raw output
[y(x) = arctanh(1/(-_C1*x^4+1)^(1/2))*x, y(x) = -arctanh(1/(-_C1*x^4+1)^(1/2))*x]