4.22.4 \(y'(x)^3+y'(x)^2 (\cos (x) \cot (x)-y(x))-y'(x) (y(x) \cos (x) \cot (x)+1)+y(x)=0\)

ODE
\[ y'(x)^3+y'(x)^2 (\cos (x) \cot (x)-y(x))-y'(x) (y(x) \cos (x) \cot (x)+1)+y(x)=0 \] ODE Classification

[_quadrature]

Book solution method
No Missing Variables ODE, Solve for \(y'\)

Mathematica ✓
cpu = 0.200277 (sec), leaf count = 45

\[\left \{\left \{y(x)\to c_1 e^x\right \},\{y(x)\to -\cos (x)+c_1\},\left \{y(x)\to -\log \left (\sin \left (\frac {x}{2}\right )\right )+\log \left (\cos \left (\frac {x}{2}\right )\right )+c_1\right \}\right \}\]

Maple ✓
cpu = 0.113 (sec), leaf count = 32

\[[y \left (x \right ) = \textit {\_C1} \,{\mathrm e}^{x}, y \left (x \right ) = -\ln \left (\csc \left (x \right )-\cot \left (x \right )\right )+\textit {\_C1}, y \left (x \right ) = -\cos \left (x \right )+\textit {\_C1}]\] Mathematica raw input

DSolve[y[x] - (1 + Cos[x]*Cot[x]*y[x])*y'[x] + (Cos[x]*Cot[x] - y[x])*y'[x]^2 + y'[x]^3 == 0,y[x],x]

Mathematica raw output

{{y[x] -> E^x*C[1]}, {y[x] -> C[1] - Cos[x]}, {y[x] -> C[1] + Log[Cos[x/2]] - Lo
g[Sin[x/2]]}}

Maple raw input

dsolve(diff(y(x),x)^3+(cos(x)*cot(x)-y(x))*diff(y(x),x)^2-(1+y(x)*cos(x)*cot(x))*diff(y(x),x)+y(x) = 0, y(x))

Maple raw output

[y(x) = _C1*exp(x), y(x) = -ln(csc(x)-cot(x))+_C1, y(x) = -cos(x)+_C1]