Internal problem ID [9005]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1426.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime } \left (\sin ^{2}\relax (x )\right )-\left (a^{2} \left (\cos ^{2}\relax (x )\right )+b \cos \relax (x )+\frac {b^{2}}{\left (2 a -3\right )^{2}}+3 a +2\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.844 (sec). Leaf size: 613
dsolve(sin(x)^2*diff(diff(y(x),x),x)-(a^2*cos(x)^2+b*cos(x)+b^2/(2*a-3)^2+3*a+2)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = \frac {c_{1} \left (\frac {\cos \relax (x )}{2}-\frac {1}{2}\right )^{\frac {4 a -6+\sqrt {16 a^{4}+\left (16 b -72\right ) a^{2}-48 b a +4 \left (b +\frac {9}{2}\right )^{2}}}{8 a -12}} \hypergeom \left (\left [\frac {8 a^{2}-\sqrt {16 a^{4}+\left (-16 b -72\right ) a^{2}+48 b a +4 \left (b -\frac {9}{2}\right )^{2}}+\sqrt {16 a^{4}+\left (16 b -72\right ) a^{2}-48 b a +4 \left (b +\frac {9}{2}\right )^{2}}-8 a -6}{8 a -12}, \frac {-8 a^{2}-\sqrt {16 a^{4}+\left (-16 b -72\right ) a^{2}+48 b a +4 \left (b -\frac {9}{2}\right )^{2}}+\sqrt {16 a^{4}+\left (16 b -72\right ) a^{2}-48 b a +4 \left (b +\frac {9}{2}\right )^{2}}+16 a -6}{8 a -12}\right ], \left [\frac {4 a -6-\sqrt {16 a^{4}+\left (-16 b -72\right ) a^{2}+48 b a +4 \left (b -\frac {9}{2}\right )^{2}}}{4 a -6}\right ], \frac {\cos \relax (x )}{2}+\frac {1}{2}\right ) \left (2 \cos \relax (x )+2\right )^{\frac {4 a -6-\sqrt {16 a^{4}+\left (-16 b -72\right ) a^{2}+48 b a +4 \left (b -\frac {9}{2}\right )^{2}}}{8 a -12}}}{\sqrt {\sin \relax (x )}}+\frac {c_{2} \left (\frac {\cos \relax (x )}{2}-\frac {1}{2}\right )^{\frac {4 a -6+\sqrt {16 a^{4}+\left (16 b -72\right ) a^{2}-48 b a +4 \left (b +\frac {9}{2}\right )^{2}}}{8 a -12}} \hypergeom \left (\left [\frac {8 a^{2}+\sqrt {16 a^{4}+\left (-16 b -72\right ) a^{2}+48 b a +4 \left (b -\frac {9}{2}\right )^{2}}+\sqrt {16 a^{4}+\left (16 b -72\right ) a^{2}-48 b a +4 \left (b +\frac {9}{2}\right )^{2}}-8 a -6}{8 a -12}, \frac {-8 a^{2}+\sqrt {16 a^{4}+\left (-16 b -72\right ) a^{2}+48 b a +4 \left (b -\frac {9}{2}\right )^{2}}+\sqrt {16 a^{4}+\left (16 b -72\right ) a^{2}-48 b a +4 \left (b +\frac {9}{2}\right )^{2}}+16 a -6}{8 a -12}\right ], \left [\frac {4 a -6+\sqrt {16 a^{4}+\left (-16 b -72\right ) a^{2}+48 b a +4 \left (b -\frac {9}{2}\right )^{2}}}{4 a -6}\right ], \frac {\cos \relax (x )}{2}+\frac {1}{2}\right ) \left (2 \cos \relax (x )+2\right )^{\frac {4 a -6+\sqrt {16 a^{4}+\left (-16 b -72\right ) a^{2}+48 b a +4 \left (b -\frac {9}{2}\right )^{2}}}{8 a -12}}}{\sqrt {\sin \relax (x )}} \]
✓ Solution by Mathematica
Time used: 2.858 (sec). Leaf size: 829
DSolve[(-2 - 3*a - b^2/(-3 + 2*a)^2 - b*Cos[x] - a^2*Cos[x]^2)*y[x] + Sin[x]^2*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {(-1)^{\frac {-4 a^2-9}{(3-2 a)^2}} 2^{-\frac {\sqrt {(2 a+3)^2 (3-2 a)^4+4 b (3-2 a)^4+4 b^2 (3-2 a)^2}}{2 (3-2 a)^2}} (\cos (x)-1)^{\frac {1}{2}-\frac {\sqrt {(2 a+3)^2 (3-2 a)^4+4 b (3-2 a)^4+4 b^2 (3-2 a)^2}}{4 (3-2 a)^2}} (\cos (x)+1)^{\frac {1}{4} \left (\sqrt {4 a (a+3)+4 b \left (\frac {b}{(3-2 a)^2}-1\right )+9}+2\right )} \left ((-1)^{\frac {4 a^2+9}{(3-2 a)^2}} 2^{\frac {\sqrt {(2 a+3)^2 (3-2 a)^4+4 b (3-2 a)^4+4 b^2 (3-2 a)^2}}{2 (3-2 a)^2}} c_1 \, _2F_1\left (\frac {1}{4} \left (-4 a-\frac {\sqrt {(3-2 a)^2 \left (4 b (3-2 a)^2+\left (9-4 a^2\right )^2+4 b^2\right )}}{(3-2 a)^2}+\sqrt {4 a (a+3)+4 b \left (\frac {b}{(3-2 a)^2}-1\right )+9}+2\right ),\frac {1}{4} \left (4 a-\frac {\sqrt {(3-2 a)^2 \left (4 b (3-2 a)^2+\left (9-4 a^2\right )^2+4 b^2\right )}}{(3-2 a)^2}+\sqrt {4 a (a+3)+4 b \left (\frac {b}{(3-2 a)^2}-1\right )+9}+2\right );1-\frac {\sqrt {(3-2 a)^2 \left (4 b (3-2 a)^2+\left (9-4 a^2\right )^2+4 b^2\right )}}{2 (3-2 a)^2};\sin ^2\left (\frac {x}{2}\right )\right )-e^{\frac {i \left (24 a+\sqrt {(2 a+3)^2 (3-2 a)^4+4 b (3-2 a)^4+4 b^2 (3-2 a)^2}\right ) \pi }{2 (3-2 a)^2}} c_2 (1-\cos (x))^{\frac {\sqrt {(2 a+3)^2 (3-2 a)^4+4 b (3-2 a)^4+4 b^2 (3-2 a)^2}}{2 (3-2 a)^2}} \, _2F_1\left (\frac {1}{4} \left (-4 a+\frac {\sqrt {(3-2 a)^2 \left (4 b (3-2 a)^2+\left (9-4 a^2\right )^2+4 b^2\right )}}{(3-2 a)^2}+\sqrt {4 a (a+3)+4 b \left (\frac {b}{(3-2 a)^2}-1\right )+9}+2\right ),\frac {1}{4} \left (4 a+\frac {\sqrt {(3-2 a)^2 \left (4 b (3-2 a)^2+\left (9-4 a^2\right )^2+4 b^2\right )}}{(3-2 a)^2}+\sqrt {4 a (a+3)+4 b \left (\frac {b}{(3-2 a)^2}-1\right )+9}+2\right );\frac {\sqrt {(3-2 a)^2 \left (4 b (3-2 a)^2+\left (9-4 a^2\right )^2+4 b^2\right )}}{2 (3-2 a)^2}+1;\sin ^2\left (\frac {x}{2}\right )\right )\right )}{\sqrt [4]{-\sin ^2(x)}} \\ \end{align*}