Internal problem ID [8666]
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]]
\[ \boxed {4 y^{\prime \prime }+4 y^{\prime } \tan \left (x \right )-\left (5 \tan \left (x \right )^{2}+2\right ) y=0} \]
✓ Solution by Maple
Time used: 0.172 (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)
\[ y \left (x \right ) = \frac {c_{1}}{\sqrt {\cos \left (x \right )}}+\frac {c_{2} \left (i \cos \left (x \right ) \sin \left (x \right )-\ln \left (i \cos \left (x \right )+\sin \left (x \right )\right )\right )}{\sqrt {\cos \left (x \right )}} \]
✓ Solution by Mathematica
Time used: 0.054 (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} \text {arcsinh}\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*}