Internal problem ID [9143]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 4, linear fourth order
Problem number: 1564.
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}+3\right ) x^{2} y^{\prime \prime }+\left (12 n^{2}-3\right ) x y^{\prime }-\left (4 x^{4}+12 n^{2}-3\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 87
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+3)*x^2*diff(diff(y(x),x),x)+(12*n^2-3)*x*diff(y(x),x)-(4*x^4+12*n^2-3)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = \frac {c_{1} \left (\mathit {ber}_{n}\relax (x )^{2}+\mathit {bei}_{n}\relax (x )^{2}\right )}{x}+\frac {c_{2} \left (\mathit {ber}_{-n}\relax (x )^{2}+\mathit {bei}_{-n}\relax (x )^{2}\right )}{x}+c_{3} x^{3} \hypergeom \left (\left [\right ], \left [\frac {3}{2}, -\frac {n}{2}+2, \frac {n}{2}+2\right ], \frac {x^{4}}{64}\right )+c_{4} x \hypergeom \left (\left [\right ], \left [\frac {1}{2}, \frac {n}{2}+\frac {3}{2}, \frac {3}{2}-\frac {n}{2}\right ], \frac {x^{4}}{64}\right ) \]
✓ Solution by Mathematica
Time used: 0.693 (sec). Leaf size: 196
DSolve[(3 - 12*n^2 - 4*x^4)*y[x] + (-3 + 12*n^2)*x*y'[x] - (3 + 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 {\left (\frac {1}{32}+\frac {i}{32}\right ) \left (i \left (c_2 x^4 \, _0F_3\left (;\frac {3}{2},2-\frac {n}{2},\frac {n}{2}+2;\frac {x^4}{64}\right )-8^{2-n} e^{-\frac {1}{2} i \pi n} x^{-2 n} \left (c_3 64^n \, _0F_3\left (;1-n,\frac {1}{2}-\frac {n}{2},-\frac {n}{2};\frac {x^4}{64}\right )+c_4 e^{i \pi n} x^{4 n} \, _0F_3\left (;\frac {n}{2}+\frac {1}{2},\frac {n}{2},n+1;\frac {x^4}{64}\right )\right )\right )+8 c_1 x^2 \, _0F_3\left (;\frac {1}{2},\frac {3}{2}-\frac {n}{2},\frac {n}{2}+\frac {3}{2};\frac {x^4}{64}\right )\right )}{x} \\ \end{align*}