4.5.9 \(x y'(x)=y(x)+x \sec ^2\left (\frac {y(x)}{x}\right )\)

ODE
\[ x y'(x)=y(x)+x \sec ^2\left (\frac {y(x)}{x}\right ) \] ODE Classification

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

Book solution method
Homogeneous equation

Mathematica
cpu = 0.0441468 (sec), leaf count = 27

\[\text {Solve}\left [4 \left (c_1+\log (x)\right )=\frac {2 y(x)}{x}+\sin \left (\frac {2 y(x)}{x}\right ),y(x)\right ]\]

Maple
cpu = 0.02 (sec), leaf count = 35

\[ \left \{ {\frac {1}{2\,x} \left (\cos \left ({\frac {y \relax (x ) }{x}} \right ) \sin \left ({\frac {y \relax (x ) }{x}} \right ) x+y \relax (x ) \right ) }-\ln \relax (x ) -{\it \_C1}=0 \right \} \] Mathematica raw input

DSolve[x*y'[x] == x*Sec[y[x]/x]^2 + y[x],y[x],x]

Mathematica raw output

Solve[4*(C[1] + Log[x]) == Sin[(2*y[x])/x] + (2*y[x])/x, y[x]]

Maple raw input

dsolve(x*diff(y(x),x) = y(x)+x*sec(y(x)/x)^2, y(x),'implicit')

Maple raw output

1/2*(cos(y(x)/x)*sin(y(x)/x)*x+y(x))/x-ln(x)-_C1 = 0