4.17.32 $y^{\prime }(x{)}^2+2y(x)\cot (x)y^{\prime }(x)-y(x{)}^2=0$

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

[_separable]

Book solution method
No Missing Variables ODE, Solve for $y^{\prime }$

Mathematica ✓
cpu = 0.032357 (sec), leaf count = 31

$$\{\{y(x)\to c_1{\csc }^2\left(\frac{x}{2}\right)\},\{y(x)\to c_1{\sec }^2\left(\frac{x}{2}\right)\}\}$$

Maple ✓
cpu = 0.107 (sec), leaf count = 91

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

DSolve[-y[x]^2 + 2*Cot[x]*y[x]*y'[x] + y'[x]^2 == 0,y[x],x]

Mathematica raw output

{{y[x] -> C[1]*Csc[x/2]^2}, {y[x] -> C[1]*Sec[x/2]^2}}

Maple raw input

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

Maple raw output

y(x)-_C1/(1/(tan(x)^2+1)^(1/2)+1)*(1-1/(tan(x)^2+1))^(1/2)*(tan(x)^2+1)^(1/2)/ta
n(x) = 0, y(x)-_C1*(1/(tan(x)^2+1)^(1/2)+1)/(1-1/(tan(x)^2+1))^(1/2)*(tan(x)^2+1
)^(1/2)/tan(x) = 0