Internal problem ID [9761]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1427.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {y^{\prime \prime }+\frac {\left (-\left (a^{2} b^{2}-\left (1+a \right )^{2}\right ) \sin \left (x \right )^{2}-a \left (1+a \right ) b \sin \left (2 x \right )-a \left (a -1\right )\right ) y}{\sin \left (x \right )^{2}}=0} \]
✓ Solution by Maple
Time used: 0.547 (sec). Leaf size: 112
dsolve(diff(diff(y(x),x),x) = -(-(a^2*b^2-(a+1)^2)*sin(x)^2-a*(a+1)*b*sin(2*x)-a*(a-1))/sin(x)^2*y(x),y(x), singsol=all)
\[ y \left (x \right ) = c_{1} \left (\cot \left (x \right )+i\right )^{-\frac {1}{2}-\frac {1}{2} i a b -\frac {1}{2} a} \left (b +\cot \left (x \right )\right ) \left (\cot \left (x \right )-i\right )^{-\frac {1}{2}+\frac {1}{2} i a b -\frac {1}{2} a}+c_{2} \left (\cot \left (x \right )+i\right )^{\frac {1}{2}+\frac {1}{2} a +\frac {1}{2} i a b} \operatorname {hypergeom}\left (\left [i a b +a , i a b -a +1\right ], \left [i a b +a +2\right ], \frac {1}{2}-\frac {i \cot \left (x \right )}{2}\right ) \left (\cot \left (x \right )-i\right )^{-\frac {1}{2}+\frac {1}{2} i a b -\frac {1}{2} a} \]
✓ Solution by Mathematica
Time used: 1.502 (sec). Leaf size: 161
DSolve[y''[x] == -(Csc[x]^2*((1 - a)*a - (-(1 + a)^2 + a^2*b^2)*Sin[x]^2 - a*(1 + a)*b*Sin[2*x])*y[x]),y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to c_2 e^{-a b x} \sin ^{-a}(x) \left (\csc (x)+\frac {2^{2 a+1} (2 a+1) e^{2 i x} \left (1-e^{2 i x}\right )^{2 a} \left (-i e^{-i x} \left (-1+e^{2 i x}\right )\right )^{-2 a} \sin ^{2 a}(x) \operatorname {Hypergeometric2F1}\left (2 a+2,i b a+a+1,i b a+a+2,e^{2 i x}\right ) (b \sin (x)+\cos (x))}{a (b-i)-i}\right )+c_1 e^{a b x} \sin ^a(x) (b \sin (x)+\cos (x)) \]