Internal problem ID [9017]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1438.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+\frac {\left (-a \left (\cos ^{2}\relax (x )\right ) \left (\sin ^{2}\relax (x )\right )-m \left (m -1\right ) \left (\sin ^{2}\relax (x )\right )-n \left (n -1\right ) \left (\cos ^{2}\relax (x )\right )\right ) y}{\cos \relax (x )^{2} \sin \relax (x )^{2}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.11 (sec). Leaf size: 105
dsolve(diff(diff(y(x),x),x) = -(-a*cos(x)^2*sin(x)^2-m*(m-1)*sin(x)^2-n*(n-1)*cos(x)^2)/cos(x)^2/sin(x)^2*y(x),y(x), singsol=all)
\[ y \relax (x ) = c_{1} \left (\cos ^{m}\relax (x )\right ) \left (\sin ^{n}\relax (x )\right ) \hypergeom \left (\left [\frac {n}{2}+\frac {m}{2}+\frac {i \sqrt {a}}{2}, \frac {n}{2}+\frac {m}{2}-\frac {i \sqrt {a}}{2}\right ], \left [\frac {1}{2}+m \right ], \cos ^{2}\relax (x )\right )+c_{2} \left (\cos ^{-m +1}\relax (x )\right ) \left (\sin ^{n}\relax (x )\right ) \hypergeom \left (\left [\frac {n}{2}-\frac {m}{2}+\frac {i \sqrt {a}}{2}+\frac {1}{2}, \frac {n}{2}-\frac {m}{2}-\frac {i \sqrt {a}}{2}+\frac {1}{2}\right ], \left [\frac {3}{2}-m \right ], \cos ^{2}\relax (x )\right ) \]
✓ Solution by Mathematica
Time used: 0.608 (sec). Leaf size: 158
DSolve[y''[x] == -(Csc[x]^2*Sec[x]^2*((1 - n)*n*Cos[x]^2 - (-1 + m)*m*Sin[x]^2 - a*Cos[x]^2*Sin[x]^2)*y[x]),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {(-1)^{-m} \cos ^2(x)^{\frac {1}{4}-\frac {m}{2}} \left (-\sin ^2(x)\right )^{n/2} \left (c_1 (-1)^m \cos ^2(x)^m \, _2F_1\left (\frac {1}{2} \left (m+n-\sqrt {-a}\right ),\frac {1}{2} \left (m+n+\sqrt {-a}\right );m+\frac {1}{2};\cos ^2(x)\right )+i c_2 \sqrt {\cos ^2(x)} \, _2F_1\left (\frac {1}{2} \left (-m+n-\sqrt {-a}+1\right ),\frac {1}{2} \left (-m+n+\sqrt {-a}+1\right );\frac {3}{2}-m;\cos ^2(x)\right )\right )}{\sqrt {\cos (x)}} \\ \end{align*}