4.3.15 $x(\sin (2y(x))-x^2{\cos }^2(y(x)))+y^{\prime }(x)=0$

ODE
$$x(\sin (2y(x))-x^2{\cos }^2(y(x)))+y^{\prime }(x)=0$$ ODE Classification

[`y=_G(x,y')`]

Book solution method
Change of Variable, new dependent variable

Mathematica ✓
cpu = 0.260147 (sec), leaf count = 55

$$\{\{y(x)\to {\tan }^{-1}\left(\frac{1}{2}(-8c_1{e}^{-x^2}+x^2-1)\right)\},\{y(x)\to -{\tan }^{-1}(4c_1{e}^{-x^2}-\frac{x^2}{2}+\frac{1}{2})\}\}$$

Maple ✓
cpu = 1.17 (sec), leaf count = 23

$$\{\ln (-x^2+2\tan \left(y\left(x\right)\right)+1)+x^2-\mathit{\_}\mathit{C}\mathit{1}=0\}$$ Mathematica raw input

DSolve[x*(-(x^2*Cos[y[x]]^2) + Sin[2*y[x]]) + y'[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> ArcTan[(-1 + x^2 - (8*C[1])/E^x^2)/2]}, {y[x] -> -ArcTan[1/2 - x^2/2 +
 (4*C[1])/E^x^2]}}

Maple raw input

dsolve(diff(y(x),x)+x*(sin(2*y(x))-x^2*cos(y(x))^2) = 0, y(x),'implicit')

Maple raw output

ln(-x^2+2*tan(y(x))+1)+x^2-_C1 = 0