4.2.44 \(y'(x)=y(x) \left (y(x)^3 \sec (x)+\tan (x)\right )\)

ODE
\[ y'(x)=y(x) \left (y(x)^3 \sec (x)+\tan (x)\right ) \] ODE Classification

[_Bernoulli]

Book solution method
The Bernoulli ODE

Mathematica
cpu = 0.0449442 (sec), leaf count = 98

\[\left \{\left \{y(x)\to \frac {1}{\sqrt [3]{c_1 \cos ^3(x)-\sin (x)-2 \sin (x) \cos ^2(x)}}\right \},\left \{y(x)\to -\frac {\sqrt [3]{-1}}{\sqrt [3]{c_1 \cos ^3(x)-\sin (x)-2 \sin (x) \cos ^2(x)}}\right \},\left \{y(x)\to \frac {(-1)^{2/3}}{\sqrt [3]{c_1 \cos ^3(x)-\sin (x)-2 \sin (x) \cos ^2(x)}}\right \}\right \}\]

Maple
cpu = 0.575 (sec), leaf count = 33

\[ \left \{ {\frac {1+ \left (- \left (\cos \relax (x ) \right ) ^{3}{\it \_C1}+2\,\sin \relax (x ) \left (\cos \relax (x ) \right ) ^{2}+\sin \relax (x ) \right ) \left (y \relax (x ) \right ) ^{3}}{ \left (y \relax (x ) \right ) ^{3}}}=0 \right \} \] Mathematica raw input

DSolve[y'[x] == y[x]*(Tan[x] + Sec[x]*y[x]^3),y[x],x]

Mathematica raw output

{{y[x] -> (C[1]*Cos[x]^3 - Sin[x] - 2*Cos[x]^2*Sin[x])^(-1/3)}, {y[x] -> -((-1)^
(1/3)/(C[1]*Cos[x]^3 - Sin[x] - 2*Cos[x]^2*Sin[x])^(1/3))}, {y[x] -> (-1)^(2/3)/
(C[1]*Cos[x]^3 - Sin[x] - 2*Cos[x]^2*Sin[x])^(1/3)}}

Maple raw input

dsolve(diff(y(x),x) = (tan(x)+y(x)^3*sec(x))*y(x), y(x),'implicit')

Maple raw output

(1+(-cos(x)^3*_C1+2*sin(x)*cos(x)^2+sin(x))*y(x)^3)/y(x)^3 = 0