Internal problem ID [9153]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 4, linear fourth order
Problem number: 1574.
ODE order: 4.
ODE degree: 1.
CAS Maple gives this as type [[_high_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime \prime \prime } \left (\sin ^{4}\relax (x )\right )+2 y^{\prime \prime \prime } \left (\sin ^{3}\relax (x )\right ) \cos \relax (x )+y^{\prime \prime } \left (\sin ^{2}\relax (x )\right ) \left (\sin ^{2}\relax (x )-3\right )+y^{\prime } \sin \relax (x ) \cos \relax (x ) \left (2 \left (\sin ^{2}\relax (x )\right )+3\right )+\left (a^{4} \left (\sin ^{4}\relax (x )\right )-3\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.25 (sec). Leaf size: 257
dsolve(diff(diff(diff(diff(y(x),x),x),x),x)*sin(x)^4+2*diff(diff(diff(y(x),x),x),x)*sin(x)^3*cos(x)+diff(diff(y(x),x),x)*sin(x)^2*(sin(x)^2-3)+diff(y(x),x)*sin(x)*cos(x)*(2*sin(x)^2+3)+(a^4*sin(x)^4-3)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} \sin \relax (x ) \hypergeom \left (\left [\frac {3}{4}+\frac {\sqrt {-4 \sqrt {-\left (a -1\right ) \left (a +1\right ) \left (a^{2}+1\right )}+5}}{4}, \frac {3}{4}-\frac {\sqrt {-4 \sqrt {-\left (a -1\right ) \left (a +1\right ) \left (a^{2}+1\right )}+5}}{4}\right ], \left [\frac {1}{2}\right ], \cos ^{2}\relax (x )\right )+c_{2} \sin \relax (x ) \hypergeom \left (\left [\frac {3}{4}+\frac {\sqrt {4 \sqrt {-\left (a -1\right ) \left (a +1\right ) \left (a^{2}+1\right )}+5}}{4}, \frac {3}{4}-\frac {\sqrt {4 \sqrt {-\left (a -1\right ) \left (a +1\right ) \left (a^{2}+1\right )}+5}}{4}\right ], \left [\frac {1}{2}\right ], \cos ^{2}\relax (x )\right )+c_{3} \sin \relax (x ) \cos \relax (x ) \hypergeom \left (\left [\frac {5}{4}+\frac {\sqrt {-4 \sqrt {-\left (a -1\right ) \left (a +1\right ) \left (a^{2}+1\right )}+5}}{4}, \frac {5}{4}-\frac {\sqrt {-4 \sqrt {-\left (a -1\right ) \left (a +1\right ) \left (a^{2}+1\right )}+5}}{4}\right ], \left [\frac {3}{2}\right ], \cos ^{2}\relax (x )\right )+c_{4} \sin \relax (x ) \cos \relax (x ) \hypergeom \left (\left [\frac {5}{4}+\frac {\sqrt {4 \sqrt {-\left (a -1\right ) \left (a +1\right ) \left (a^{2}+1\right )}+5}}{4}, \frac {5}{4}-\frac {\sqrt {4 \sqrt {-\left (a -1\right ) \left (a +1\right ) \left (a^{2}+1\right )}+5}}{4}\right ], \left [\frac {3}{2}\right ], \cos ^{2}\relax (x )\right ) \]
✓ Solution by Mathematica
Time used: 0.119 (sec). Leaf size: 265
DSolve[(-3 + a^4*Sin[x]^4)*y[x] + Cos[x]*Sin[x]*(3 + 2*Sin[x]^2)*y'[x] + Sin[x]^2*(-3 + Sin[x]^2)*y''[x] + 2*Cos[x]*Sin[x]^3*Derivative[3][y][x] + Sin[x]^4*Derivative[4][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \sin (x) \left (c_1 \, _2F_1\left (\frac {1}{4} \left (3-\sqrt {5-4 \sqrt {1-a^4}}\right ),\frac {1}{4} \left (\sqrt {5-4 \sqrt {1-a^4}}+3\right );\frac {1}{2};\cos ^2(x)\right )+c_3 \cos (x) \, _2F_1\left (\frac {1}{4} \left (5-\sqrt {5-4 \sqrt {1-a^4}}\right ),\frac {1}{4} \left (\sqrt {5-4 \sqrt {1-a^4}}+5\right );\frac {3}{2};\cos ^2(x)\right )+c_2 \, _2F_1\left (\frac {1}{4} \left (3-\sqrt {4 \sqrt {1-a^4}+5}\right ),\frac {1}{4} \left (\sqrt {4 \sqrt {1-a^4}+5}+3\right );\frac {1}{2};\cos ^2(x)\right )+c_4 \cos (x) \, _2F_1\left (\frac {1}{4} \left (5-\sqrt {4 \sqrt {1-a^4}+5}\right ),\frac {1}{4} \left (\sqrt {4 \sqrt {1-a^4}+5}+5\right );\frac {3}{2};\cos ^2(x)\right )\right ) \\ \end{align*}