Internal problem ID [9140]
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]]
Solve \begin {gather*} \boxed {x^{4} y^{\prime \prime \prime \prime }-2 n \left (n +1\right ) x^{2} y^{\prime \prime }+4 n \left (n +1\right ) x y^{\prime }+\left (a \,x^{4}+n \left (n +1\right ) \left (n +3\right ) \left (n -2\right )\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.02 (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)
\[ y \relax (x ) = c_{1} \sqrt {x}\, \BesselJ \left (n +\frac {1}{2}, \left (-a \right )^{\frac {1}{4}} x \right )+c_{2} \sqrt {x}\, \BesselY \left (n +\frac {1}{2}, \left (-a \right )^{\frac {1}{4}} x \right )+c_{3} \sqrt {x}\, \BesselJ \left (n +\frac {1}{2}, \sqrt {-\sqrt {-a}}\, x \right )+c_{4} \sqrt {x}\, \BesselY \left (n +\frac {1}{2}, \sqrt {-\sqrt {-a}}\, x \right ) \]
✓ Solution by Mathematica
Time used: 2.075 (sec). Leaf size: 252
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]
\begin{align*} y(x)\to \sqrt [8]{a} 2^{-n-\frac {9}{2}} \sqrt {x} \left (2^{2 n+1} \Gamma \left (\frac {1}{2}-n\right ) \left (\text {ber}_{-n-\frac {1}{2}}\left (\sqrt [4]{a} x\right ) \left (8 c_2 \cos \left (\frac {3}{8} (2 \pi n+\pi )\right )+c_1 (1-2 n) \sin \left (\frac {3}{8} (2 \pi n+\pi )\right )\right )+\text {bei}_{-n-\frac {1}{2}}\left (\sqrt [4]{a} x\right ) \left (c_1 (1-2 n) \cos \left (\frac {3}{8} (2 \pi n+\pi )\right )-8 c_2 \sin \left (\frac {3}{8} (2 \pi n+\pi )\right )\right )\right )+\Gamma \left (n+\frac {3}{2}\right ) \left (\text {ber}_{n+\frac {1}{2}}\left (\sqrt [4]{a} x\right ) \left (8 c_3 \cos \left (\frac {3}{8} (2 \pi n+\pi )\right )-c_4 (2 n+3) \sin \left (\frac {3}{8} (2 \pi n+\pi )\right )\right )+\text {bei}_{n+\frac {1}{2}}\left (\sqrt [4]{a} x\right ) \left (c_4 (2 n+3) \cos \left (\frac {3}{8} (2 \pi n+\pi )\right )+8 c_3 \sin \left (\frac {3}{8} (2 \pi n+\pi )\right )\right )\right )\right ) \\ \end{align*}