##### 4.25.25 $$y''(x)=y(x) \left (2 \tan ^2(x)+1\right )$$

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 = 0.320016 (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 = 0.6 (sec), leaf count = 34

$\left [y \left (x \right ) = \frac {\textit {\_C1}}{\cos \left (x \right )}+\frac {\textit {\_C2} \left (i \cos \left (x \right ) \sin \left (x \right )+\ln \left (\cos \left (x \right )+i \sin \left (x \right )\right )\right )}{\cos \left (x \right )}\right ]$ 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]*S
qrt[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))

Maple raw output

[y(x) = _C1/cos(x)+_C2/cos(x)*(I*cos(x)*sin(x)+ln(cos(x)+I*sin(x)))]