ODE
\[ y(x) y'(x)=\csc ^2(x)-y(x)^2 \cot (x) \] ODE Classification
[_Bernoulli]
Book solution method
The Bernoulli ODE
Mathematica ✓
cpu = 0.502183 (sec), leaf count = 36
\[\left \{\left \{y(x)\to -\sqrt {2 x+c_1} \csc (x)\right \},\left \{y(x)\to \sqrt {2 x+c_1} \csc (x)\right \}\right \}\]
Maple ✓
cpu = 0.056 (sec), leaf count = 32
\[\left [y \left (x \right ) = \frac {\sqrt {2 x +\textit {\_C1}}}{\sin \left (x \right )}, y \left (x \right ) = -\frac {\sqrt {2 x +\textit {\_C1}}}{\sin \left (x \right )}\right ]\] Mathematica raw input
DSolve[y[x]*y'[x] == Csc[x]^2 - Cot[x]*y[x]^2,y[x],x]
Mathematica raw output
{{y[x] -> -(Sqrt[2*x + C[1]]*Csc[x])}, {y[x] -> Sqrt[2*x + C[1]]*Csc[x]}}
Maple raw input
dsolve(y(x)*diff(y(x),x) = csc(x)^2-y(x)^2*cot(x), y(x))
Maple raw output
[y(x) = (2*x+_C1)^(1/2)/sin(x), y(x) = -(2*x+_C1)^(1/2)/sin(x)]