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 = 0.032357 (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 = 0.107 (sec), leaf count = 91

\[ \left \{ y \relax (x ) -{\frac {{\it \_C1}}{\tan \relax (x ) }\sqrt {1- \left (\left (\tan \relax (x ) \right ) ^{2}+1 \right ) ^{-1}}\sqrt { \left (\tan \relax (x ) \right ) ^{2}+1} \left ({\frac {1}{\sqrt { \left (\tan \relax (x ) \right ) ^{2}+1}}}+1 \right ) ^{-1}}=0,y \relax (x ) -{\frac {{\it \_C1}}{\tan \relax (x ) } \left ({\frac {1}{\sqrt { \left (\tan \relax (x ) \right ) ^{2}+1}}}+1 \right ) \sqrt { \left (\tan \relax (x ) \right ) ^{2}+1}{\frac {1}{\sqrt {1- \left (\left (\tan \relax (x ) \right ) ^{2}+1 \right ) ^{-1}}}}}=0 \right \} \] 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),'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