Internal problem ID [9105]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 3, linear third order
Problem number: 1530.
ODE order: 3.
ODE degree: 1.
CAS Maple gives this as type [[_3rd_order, _with_linear_symmetries]]
\[ \boxed {y^{\prime \prime \prime } \sin \left (x \right )^{2}+3 y^{\prime \prime } \sin \left (x \right ) \cos \left (x \right )+\left (\cos \left (2 x \right )+4 \nu \left (\nu +1\right ) \sin \left (x \right )^{2}\right ) y^{\prime }+2 \nu \left (\nu +1\right ) y \sin \left (2 x \right )=0} \]
✓ Solution by Maple
Time used: 0.297 (sec). Leaf size: 113
dsolve(diff(diff(diff(y(x),x),x),x)*sin(x)^2+3*diff(diff(y(x),x),x)*sin(x)*cos(x)+(cos(2*x)+4*nu*(nu+1)*sin(x)^2)*diff(y(x),x)+2*nu*(nu+1)*y(x)*sin(2*x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} \operatorname {hypergeom}\left (\left [-\frac {\nu }{2}, \frac {\nu }{2}+\frac {1}{2}\right ], \left [\frac {1}{2}\right ], \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right )^{2}+c_{2} \left (1+\cos \left (2 x \right )\right ) \operatorname {hypergeom}\left (\left [\frac {\nu }{2}+1, \frac {1}{2}-\frac {\nu }{2}\right ], \left [\frac {3}{2}\right ], \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right )^{2}+c_{3} \operatorname {hypergeom}\left (\left [-\frac {\nu }{2}, \frac {\nu }{2}+\frac {1}{2}\right ], \left [\frac {1}{2}\right ], \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right ) \sqrt {1+\cos \left (2 x \right )}\, \operatorname {hypergeom}\left (\left [\frac {\nu }{2}+1, \frac {1}{2}-\frac {\nu }{2}\right ], \left [\frac {3}{2}\right ], \frac {\cos \left (2 x \right )}{2}+\frac {1}{2}\right ) \]
✓ Solution by Mathematica
Time used: 0.054 (sec). Leaf size: 35
DSolve[2*nu*(1 + nu)*Sin[2*x]*y[x] + (Cos[2*x] + 4*nu*(1 + nu)*Sin[x]^2)*y'[x] + 3*Cos[x]*Sin[x]*y''[x] + Sin[x]^2*Derivative[3][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_3 \operatorname {LegendreP}(\nu ,\cos (x)) \operatorname {LegendreQ}(\nu ,\cos (x))+c_1 \operatorname {LegendreP}(\nu ,\cos (x))^2+c_2 \operatorname {LegendreQ}(\nu ,\cos (x))^2 \\ \end{align*}