Internal problem ID [9142]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 4, linear fourth order
Problem number: 1563.
ODE order: 4.
ODE degree: 1.
CAS Maple gives this as type [[_high_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x^{4} y^{\prime \prime \prime \prime }+4 y^{\prime \prime \prime } x^{3}-\left (4 n^{2}-1\right ) x^{2} y^{\prime \prime }-\left (4 n^{2}-1\right ) x y^{\prime }+\left (-4 x^{4}+4 n^{2}-1\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.02 (sec). Leaf size: 83
dsolve(x^4*diff(diff(diff(diff(y(x),x),x),x),x)+4*x^3*diff(diff(diff(y(x),x),x),x)-(4*n^2-1)*x^2*diff(diff(y(x),x),x)-(4*n^2-1)*x*diff(y(x),x)+(-4*x^4+4*n^2-1)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} x \left (\mathit {ber}_{n}\relax (x )^{2}+\mathit {bei}_{n}\relax (x )^{2}\right )+c_{2} x \left (\mathit {ber}_{-n}\relax (x )^{2}+\mathit {bei}_{-n}\relax (x )^{2}\right )+c_{3} x \hypergeom \left (\left [\right ], \left [\frac {3}{2}, 1+\frac {n}{2}, -\frac {n}{2}+1\right ], \frac {x^{4}}{64}\right )+\frac {c_{4} \hypergeom \left (\left [\right ], \left [\frac {1}{2}, \frac {n}{2}+\frac {1}{2}, -\frac {n}{2}+\frac {1}{2}\right ], \frac {x^{4}}{64}\right )}{x} \]
✓ Solution by Mathematica
Time used: 1.056 (sec). Leaf size: 187
DSolve[(-1 + 4*n^2 - 4*x^4)*y[x] - (-1 + 4*n^2)*x*y'[x] - (-1 + 4*n^2)*x^2*y''[x] + 4*x^3*Derivative[3][y][x] + x^4*Derivative[4][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\sqrt [4]{-1} \left (x^2 \left (c_2 \, _0F_3\left (;\frac {3}{2},1-\frac {n}{2},\frac {n}{2}+1;\frac {x^4}{64}\right )+c_3 \left (\frac {i}{8}\right )^{-n} x^{-2 n} \, _0F_3\left (;1-n,1-\frac {n}{2},\frac {3}{2}-\frac {n}{2};\frac {x^4}{64}\right )+c_4 \left (\frac {i}{8}\right )^n x^{2 n} \, _0F_3\left (;\frac {n}{2}+1,\frac {n}{2}+\frac {3}{2},n+1;\frac {x^4}{64}\right )\right )-8 i c_1 \, _0F_3\left (;\frac {1}{2},\frac {1}{2}-\frac {n}{2},\frac {n}{2}+\frac {1}{2};\frac {x^4}{64}\right )\right )}{2 \sqrt {2} x} \\ \end{align*}