Internal problem ID [9006]
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]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+\frac {\left (-\left (a^{2} b^{2}-\left (a +1\right )^{2}\right ) \left (\sin ^{2}\relax (x )\right )-a \left (a +1\right ) b \sin \left (2 x \right )-a \left (a -1\right )\right ) y}{\sin \relax (x )^{2}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.344 (sec). Leaf size: 262
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)
\[ y \relax (x ) = \frac {c_{1} {\mathrm e}^{\int \frac {2 b \left (\left (a +1\right ) \cos \left (2 x \right )+a +\frac {1}{2}\right ) \sin \left (2 x \right )-\left (\cos \left (2 x \right )+1\right ) \left (\left (a \,b^{2}-a -2\right ) \cos \left (2 x \right )-a \,b^{2}-a +1\right )}{\sin \left (2 x \right ) \left (b \sin \left (2 x \right )+\cos \left (2 x \right )+1\right )}d x}}{\sqrt {\sin \left (2 x \right )}}+\frac {c_{2} {\mathrm e}^{\int \frac {2 b \left (\left (a +1\right ) \cos \left (2 x \right )+a +\frac {1}{2}\right ) \sin \left (2 x \right )-\left (\cos \left (2 x \right )+1\right ) \left (\left (a \,b^{2}-a -2\right ) \cos \left (2 x \right )-a \,b^{2}-a +1\right )}{\sin \left (2 x \right ) \left (b \sin \left (2 x \right )+\cos \left (2 x \right )+1\right )}d x} \left (\int -2 \,{\mathrm e}^{-2 \left (\int \frac {2 b \left (\left (a +1\right ) \cos \left (2 x \right )+a +\frac {1}{2}\right ) \sin \left (2 x \right )-\left (\cos \left (2 x \right )+1\right ) \left (\left (a \,b^{2}-a -2\right ) \cos \left (2 x \right )-a \,b^{2}-a +1\right )}{\sin \left (2 x \right ) \left (b \sin \left (2 x \right )+\cos \left (2 x \right )+1\right )}d x \right )} \sin \left (2 x \right )d x \right )}{\sqrt {\sin \left (2 x \right )}} \]
✓ Solution by Mathematica
Time used: 0.564 (sec). Leaf size: 106
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]
\begin{align*} y(x)\to e^{-a b x} \sin ^{-a}(x) \left (-(2 a+1) c_2 \sin (x) (\cot (x)+i) \Gamma (i b a+a+1) (b+\cot (x)) \, _2\tilde {F}_1\left (1,i a (b+i);i b a+a+2;e^{2 i x}\right )+c_1 e^{2 a b x} \sin ^{2 a+1}(x) (b+\cot (x))+c_2 \csc (x)\right ) \\ \end{align*}