ODE
\[ y''(x)=y(x) \left (2 \tan ^2(x)+1\right ) \] ODE Classification
[[_2nd_order, _with_linear_symmetries]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.153532 (sec), leaf count = 54
\[\left \{\left \{y(x)\to \frac {\sqrt [4]{\sin ^2(x)} \left (-c_2 \sqrt {\sin ^2(x)}+2 c_1 \sec (x)+c_2 \sec (x) \sin ^{-1}(\cos (x))\right )}{2 \sqrt [4]{-\sin ^2(x)}}\right \}\right \}\]
Maple ✓
cpu = 0.15 (sec), leaf count = 30
\[ \left \{ y \left ( x \right ) ={\frac {i\sin \left ( x \right ) \cos \left ( x \right ) {\it \_C2}+\ln \left ( \cos \left ( x \right ) +i\sin \left ( x \right ) \right ) {\it \_C2}+{\it \_C1}}{\cos \left ( x \right ) }} \right \} \] Mathematica raw input
DSolve[y''[x] == (1 + 2*Tan[x]^2)*y[x],y[x],x]
Mathematica raw output
{{y[x] -> ((Sin[x]^2)^(1/4)*(2*C[1]*Sec[x] + ArcSin[Cos[x]]*C[2]*Sec[x] - C[2]*Sqrt[Sin[x]^2]))/(2*(-Sin[x]^2)^(1/4))}}
Maple raw input
dsolve(diff(diff(y(x),x),x) = (1+2*tan(x)^2)*y(x), y(x),'implicit')
Maple raw output
y(x) = (I*sin(x)*cos(x)*_C2+ln(cos(x)+I*sin(x))*_C2+_C1)/cos(x)