Internal problem ID [8668]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1088.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {4 y^{\prime \prime }+4 \tan \relax (x ) y^{\prime }-\left (5 \left (\tan ^{2}\relax (x )\right )+2\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.062 (sec). Leaf size: 36
dsolve(4*diff(diff(y(x),x),x)+4*diff(y(x),x)*tan(x)-(5*tan(x)^2+2)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = \frac {c_{1}}{\sqrt {\cos \relax (x )}}+\frac {c_{2} \left (i \cos \relax (x ) \sin \relax (x )-\ln \left (i \cos \relax (x )+\sin \relax (x )\right )\right )}{\sqrt {\cos \relax (x )}} \]
✓ Solution by Mathematica
Time used: 0.048 (sec). Leaf size: 93
DSolve[(-2 - 5*Tan[x]^2)*y[x] + 4*Tan[x]*y'[x] + 4*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {3 \sqrt [8]{-1} c_2 \left ((-1)^{3/4} \sinh ^{-1}\left (\sqrt [4]{-1} \sqrt [4]{-\cos ^4(x)}\right )+\sqrt [4]{-\cos ^4(x)} \sqrt {1+i \sqrt {-\cos ^4(x)}}\right )-2 (-1)^{7/8} c_1}{2 \sqrt [8]{-\cos ^4(x)}} \\ \end{align*}