Internal problem ID [8604]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1024.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime }-\left (1+2 \left (\tan ^{2}\relax (x )\right )\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.051 (sec). Leaf size: 34
dsolve(diff(diff(y(x),x),x)-(1+2*tan(x)^2)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = \frac {c_{1}}{\cos \relax (x )}+\frac {c_{2} \left (i \cos \relax (x ) \sin \relax (x )+\ln \left (\cos \relax (x )+i \sin \relax (x )\right )\right )}{\cos \relax (x )} \]
✓ Solution by Mathematica
Time used: 0.135 (sec). Leaf size: 66
DSolve[(-1 - 2*Tan[x]^2)*y[x] + y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\sqrt [4]{\sin ^2(x)} \left (-c_2 \sqrt {\sin ^2(x)}+2 c_1 \sec (x)+2 c_2 \sec (x) \cot ^{-1}\left (\left (\sqrt {\sin ^2(x)}-1\right ) \sec (x)\right )\right )}{2 \sqrt [4]{-\sin ^2(x)}} \\ \end{align*}