Internal problem ID [8602]
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]]
\[ \boxed {y^{\prime \prime }-\left (1+2 \tan \left (x \right )^{2}\right ) y=0} \]
✓ Solution by Maple
Time used: 0.157 (sec). Leaf size: 30
dsolve(diff(diff(y(x),x),x)-(1+2*tan(x)^2)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} \sec \left (x \right )+c_{2} \sec \left (x \right ) \left (i \cos \left (x \right ) \sin \left (x \right )+\ln \left (\cos \left (x \right )+i \sin \left (x \right )\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.146 (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*}