Internal problem ID [9405]
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]]
\[ \boxed {y^{\prime \prime }+a y^{\prime } \tan \left (x \right )+b y=0} \]
✓ Solution by Maple
Time used: 0.188 (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)
\[ y \left (x \right ) = c_{1} \cos \left (x \right )^{\frac {1}{2}+\frac {a}{2}} \operatorname {LegendreP}\left (\frac {\sqrt {a^{2}+4 b}}{2}-\frac {1}{2}, \frac {1}{2}+\frac {a}{2}, \sin \left (x \right )\right )+c_{2} \cos \left (x \right )^{\frac {1}{2}+\frac {a}{2}} \operatorname {LegendreQ}\left (\frac {\sqrt {a^{2}+4 b}}{2}-\frac {1}{2}, \frac {1}{2}+\frac {a}{2}, \sin \left (x \right )\right ) \]
✓ Solution by Mathematica
Time used: 0.42 (sec). Leaf size: 129
DSolve[b*y[x] + a*Tan[x]*y'[x] + y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to c_1 \operatorname {Hypergeometric2F1}\left (\frac {1}{4} \left (-a-\sqrt {a^2+4 b}\right ),\frac {1}{4} \left (\sqrt {a^2+4 b}-a\right ),\frac {1-a}{2},\cos ^2(x)\right )+i^{a+1} c_2 \cos ^{a+1}(x) \operatorname {Hypergeometric2F1}\left (\frac {1}{4} \left (a-\sqrt {a^2+4 b}+2\right ),\frac {1}{4} \left (a+\sqrt {a^2+4 b}+2\right ),\frac {a+3}{2},\cos ^2(x)\right ) \]