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 = 0.266836 (sec), leaf count = 98
\[\left \{\left \{y(x)\to \frac {1}{\sqrt [3]{-\sin (x)+c_1 \cos ^3(x)-2 \sin (x) \cos ^2(x)}}\right \},\left \{y(x)\to -\frac {\sqrt [3]{-1}}{\sqrt [3]{-\sin (x)+c_1 \cos ^3(x)-2 \sin (x) \cos ^2(x)}}\right \},\left \{y(x)\to \frac {(-1)^{2/3}}{\sqrt [3]{-\sin (x)+c_1 \cos ^3(x)-2 \sin (x) \cos ^2(x)}}\right \}\right \}\]
Maple ✓
cpu = 0.133 (sec), leaf count = 364
\[\left [y \left (x \right ) = \frac {\left (\cos \left (x \right ) \left (\textit {\_C1} \left (\sin ^{4}\left (x \right )\right )+2 \cos \left (x \right ) \left (\sin ^{3}\left (x \right )\right )-2 \textit {\_C1} \left (\sin ^{2}\left (x \right )\right )-3 \cos \left (x \right ) \sin \left (x \right )+\textit {\_C1} \right )^{2}\right )^{\frac {1}{3}}}{\textit {\_C1} \left (\sin ^{4}\left (x \right )\right )+2 \cos \left (x \right ) \left (\sin ^{3}\left (x \right )\right )-2 \textit {\_C1} \left (\sin ^{2}\left (x \right )\right )-3 \cos \left (x \right ) \sin \left (x \right )+\textit {\_C1}}, y \left (x \right ) = -\frac {\left (\cos \left (x \right ) \left (\textit {\_C1} \left (\sin ^{4}\left (x \right )\right )+2 \cos \left (x \right ) \left (\sin ^{3}\left (x \right )\right )-2 \textit {\_C1} \left (\sin ^{2}\left (x \right )\right )-3 \cos \left (x \right ) \sin \left (x \right )+\textit {\_C1} \right )^{2}\right )^{\frac {1}{3}}}{2 \left (\textit {\_C1} \left (\sin ^{4}\left (x \right )\right )+2 \cos \left (x \right ) \left (\sin ^{3}\left (x \right )\right )-2 \textit {\_C1} \left (\sin ^{2}\left (x \right )\right )-3 \cos \left (x \right ) \sin \left (x \right )+\textit {\_C1} \right )}-\frac {i \sqrt {3}\, \left (\cos \left (x \right ) \left (\textit {\_C1} \left (\sin ^{4}\left (x \right )\right )+2 \cos \left (x \right ) \left (\sin ^{3}\left (x \right )\right )-2 \textit {\_C1} \left (\sin ^{2}\left (x \right )\right )-3 \cos \left (x \right ) \sin \left (x \right )+\textit {\_C1} \right )^{2}\right )^{\frac {1}{3}}}{2 \left (\textit {\_C1} \left (\sin ^{4}\left (x \right )\right )+2 \cos \left (x \right ) \left (\sin ^{3}\left (x \right )\right )-2 \textit {\_C1} \left (\sin ^{2}\left (x \right )\right )-3 \cos \left (x \right ) \sin \left (x \right )+\textit {\_C1} \right )}, y \left (x \right ) = -\frac {\left (\cos \left (x \right ) \left (\textit {\_C1} \left (\sin ^{4}\left (x \right )\right )+2 \cos \left (x \right ) \left (\sin ^{3}\left (x \right )\right )-2 \textit {\_C1} \left (\sin ^{2}\left (x \right )\right )-3 \cos \left (x \right ) \sin \left (x \right )+\textit {\_C1} \right )^{2}\right )^{\frac {1}{3}}}{2 \left (\textit {\_C1} \left (\sin ^{4}\left (x \right )\right )+2 \cos \left (x \right ) \left (\sin ^{3}\left (x \right )\right )-2 \textit {\_C1} \left (\sin ^{2}\left (x \right )\right )-3 \cos \left (x \right ) \sin \left (x \right )+\textit {\_C1} \right )}+\frac {i \sqrt {3}\, \left (\cos \left (x \right ) \left (\textit {\_C1} \left (\sin ^{4}\left (x \right )\right )+2 \cos \left (x \right ) \left (\sin ^{3}\left (x \right )\right )-2 \textit {\_C1} \left (\sin ^{2}\left (x \right )\right )-3 \cos \left (x \right ) \sin \left (x \right )+\textit {\_C1} \right )^{2}\right )^{\frac {1}{3}}}{2 \textit {\_C1} \left (\sin ^{4}\left (x \right )\right )+4 \cos \left (x \right ) \left (\sin ^{3}\left (x \right )\right )-4 \textit {\_C1} \left (\sin ^{2}\left (x \right )\right )-6 \cos \left (x \right ) \sin \left (x \right )+2 \textit {\_C1}}\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))
Maple raw output
[y(x) = 1/(_C1*sin(x)^4+2*cos(x)*sin(x)^3-2*_C1*sin(x)^2-3*cos(x)*sin(x)+_C1)*(cos(x)*(_C1*sin(x)^4+2*cos(x)*sin(x)^3-2*_C1*sin(x)^2-3*cos(x)*sin(x)+_C1)^2)^(1/3), y(x) = -1/2/(_C1*sin(x)^4+2*cos(x)*sin(x)^3-2*_C1*sin(x)^2-3*cos(x)*sin(x)+_C1)*(cos(x)*(_C1*sin(x)^4+2*cos(x)*sin(x)^3-2*_C1*sin(x)^2-3*cos(x)*sin(x)+_C1)^2)^(1/3)-1/2*I*3^(1/2)/(_C1*sin(x)^4+2*cos(x)*sin(x)^3-2*_C1*sin(x)^2-3*cos(x)*sin(x)+_C1)*(cos(x)*(_C1*sin(x)^4+2*cos(x)*sin(x)^3-2*_C1*sin(x)^2-3*cos(x)*sin(x)+_C1)^2)^(1/3), y(x) = -1/2/(_C1*sin(x)^4+2*cos(x)*sin(x)^3-2*_C1*sin(x)^2-3*cos(x)*sin(x)+_C1)*(cos(x)*(_C1*sin(x)^4+2*cos(x)*sin(x)^3-2*_C1*sin(x)^2-3*cos(x)*sin(x)+_C1)^2)^(1/3)+1/2*I*3^(1/2)/(_C1*sin(x)^4+2*cos(x)*sin(x)^3-2*_C1*sin(x)^2-3*cos(x)*sin(x)+_C1)*(cos(x)*(_C1*sin(x)^4+2*cos(x)*sin(x)^3-2*_C1*sin(x)^2-3*cos(x)*sin(x)+_C1)^2)^(1/3)]