Internal problem ID [9893]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 4, linear fourth order
Problem number: 1561.
ODE order: 4.
ODE degree: 1.
CAS Maple gives this as type [[_high_order, _with_linear_symmetries]]
\[ \boxed {x^{4} y^{\prime \prime \prime \prime }-2 n \left (1+n \right ) x^{2} y^{\prime \prime }+4 n \left (1+n \right ) x y^{\prime }+\left (a \,x^{4}+n \left (1+n \right ) \left (n +3\right ) \left (-2+n \right )\right ) y=0} \]
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 77
dsolve(x^4*diff(diff(diff(diff(y(x),x),x),x),x)-2*n*(n+1)*x^2*diff(diff(y(x),x),x)+4*n*(n+1)*x*diff(y(x),x)+(a*x^4+n*(n+1)*(n+3)*(n-2))*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} \sqrt {x}\, \operatorname {BesselJ}\left (n +\frac {1}{2}, \left (-a \right )^{\frac {1}{4}} x \right )+c_{2} \sqrt {x}\, \operatorname {BesselY}\left (n +\frac {1}{2}, \left (-a \right )^{\frac {1}{4}} x \right )+c_{3} \sqrt {x}\, \operatorname {BesselJ}\left (n +\frac {1}{2}, \sqrt {-\sqrt {-a}}\, x \right )+c_{4} \sqrt {x}\, \operatorname {BesselY}\left (n +\frac {1}{2}, \sqrt {-\sqrt {-a}}\, x \right ) \]
✓ Solution by Mathematica
Time used: 3.44 (sec). Leaf size: 310
DSolve[((-2 + n)*n*(1 + n)*(3 + n) + a*x^4)*y[x] + 4*n*(1 + n)*x*y'[x] - 2*n*(1 + n)*x^2*y''[x] + x^4*Derivative[4][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \sqrt [8]{a} 2^{-n-\frac {7}{2}} \sqrt {x} \left (2^{2 n+1} \text {ber}_{-n-\frac {1}{2}}\left (\sqrt [4]{a} x\right ) \left (4 c_2 \cos \left (\frac {3}{8} \pi (2 n+1)\right ) \operatorname {Gamma}\left (\frac {1}{2}-n\right )-c_1 \cos \left (\frac {3}{8} \pi (2 n-3)\right ) \operatorname {Gamma}\left (\frac {3}{2}-n\right )\right )+\text {ber}_{n+\frac {1}{2}}\left (\sqrt [4]{a} x\right ) \left (4 c_3 \cos \left (\frac {3}{8} \pi (2 n+1)\right ) \operatorname {Gamma}\left (n+\frac {3}{2}\right )-c_4 \cos \left (\frac {3}{8} \pi (2 n+5)\right ) \operatorname {Gamma}\left (n+\frac {5}{2}\right )\right )+c_1 2^{2 n+1} \sin \left (\frac {3}{8} \pi (2 n-3)\right ) \operatorname {Gamma}\left (\frac {3}{2}-n\right ) \text {bei}_{-n-\frac {1}{2}}\left (\sqrt [4]{a} x\right )-c_2 2^{2 n+3} \sin \left (\frac {3}{8} \pi (2 n+1)\right ) \operatorname {Gamma}\left (\frac {1}{2}-n\right ) \text {bei}_{-n-\frac {1}{2}}\left (\sqrt [4]{a} x\right )+4 c_3 \sin \left (\frac {3}{8} \pi (2 n+1)\right ) \operatorname {Gamma}\left (n+\frac {3}{2}\right ) \text {bei}_{n+\frac {1}{2}}\left (\sqrt [4]{a} x\right )-c_4 \sin \left (\frac {3}{8} \pi (2 n+5)\right ) \operatorname {Gamma}\left (n+\frac {5}{2}\right ) \text {bei}_{n+\frac {1}{2}}\left (\sqrt [4]{a} x\right )\right ) \]