##### 4.17.32 $$y'(x)^2+2 y(x) \cot (x) y'(x)-y(x)^2=0$$

ODE
$y'(x)^2+2 y(x) \cot (x) y'(x)-y(x)^2=0$ ODE Classiﬁcation

[_separable]

Book solution method
No Missing Variables ODE, Solve for $$y'$$

Mathematica
cpu = 0.271835 (sec), leaf count = 31

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

Maple
cpu = 0.771 (sec), leaf count = 85

$\left [y \left (x \right ) = \frac {\textit {\_C1} \sqrt {1-\frac {1}{\tan ^{2}\left (x \right )+1}}\, \sqrt {\tan ^{2}\left (x \right )+1}}{\left (\frac {1}{\sqrt {\tan ^{2}\left (x \right )+1}}+1\right ) \tan \left (x \right )}, y \left (x \right ) = \frac {\textit {\_C1} \left (\frac {1}{\sqrt {\tan ^{2}\left (x \right )+1}}+1\right ) \sqrt {\tan ^{2}\left (x \right )+1}}{\sqrt {1-\frac {1}{\tan ^{2}\left (x \right )+1}}\, \tan \left (x \right )}\right ]$ 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))

Maple raw output

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