Internal problem ID [8650]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1070.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+a y^{\prime } \tan \relax (x )+b y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 67
dsolve(diff(diff(y(x),x),x)+a*diff(y(x),x)*tan(x)+b*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} \left (\cos ^{\frac {1}{2}+\frac {a}{2}}\relax (x )\right ) \LegendreP \left (\frac {\sqrt {a^{2}+4 b}}{2}-\frac {1}{2}, \frac {1}{2}+\frac {a}{2}, \sin \relax (x )\right )+c_{2} \left (\cos ^{\frac {1}{2}+\frac {a}{2}}\relax (x )\right ) \LegendreQ \left (\frac {\sqrt {a^{2}+4 b}}{2}-\frac {1}{2}, \frac {1}{2}+\frac {a}{2}, \sin \relax (x )\right ) \]
✓ Solution by Mathematica
Time used: 0.178 (sec). Leaf size: 129
DSolve[b*y[x] + a*Tan[x]*y'[x] + y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_1 \, _2F_1\left (\frac {1}{4} \left (-a-\sqrt {a^2+4 b}\right ),\frac {1}{4} \left (\sqrt {a^2+4 b}-a\right );\frac {1-a}{2};\cos ^2(x)\right )+i^{a+1} c_2 \cos ^{a+1}(x) \, _2F_1\left (\frac {1}{4} \left (a-\sqrt {a^2+4 b}+2\right ),\frac {1}{4} \left (a+\sqrt {a^2+4 b}+2\right );\frac {a+3}{2};\cos ^2(x)\right ) \\ \end{align*}