54.5.24 problem 1561

Internal problem ID [12820]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 4, linear fourth order
Problem number : 1561
Date solved : Friday, October 03, 2025 at 03:47:29 AM
CAS classification : [[_high_order, _with_linear_symmetries]]

\begin{align*} 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{align*}

✓ Maple. Time used: 0.020 (sec). Leaf size: 69

ode:=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)*(3+n)*(n-2))*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \sqrt {x}\, \left (\operatorname {BesselJ}\left (n +\frac {1}{2}, \left (-a \right )^{{1}/{4}} x \right ) c_1 +\operatorname {BesselJ}\left (n +\frac {1}{2}, \sqrt {-\sqrt {-a}}\, x \right ) c_3 +\operatorname {BesselY}\left (n +\frac {1}{2}, \left (-a \right )^{{1}/{4}} x \right ) c_2 +\operatorname {BesselY}\left (n +\frac {1}{2}, \sqrt {-\sqrt {-a}}\, x \right ) c_4 \right ) \]

✓ Mathematica. Time used: 1.103 (sec). Leaf size: 310

ode=((-2 + n)*n*(1 + n)*(3 + n) + a*x^4)*y[x] + 4*n*(1 + n)*x*D[y[x],x] - 2*n*(1 + n)*x^2*D[y[x],{x,2}] + x^4*Derivative[4][y][x] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} 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 ) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
n = symbols("n") 
y = Function("y") 
ode = Eq(-2*n*x**2*(n + 1)*Derivative(y(x), (x, 2)) + 4*n*x*(n + 1)*Derivative(y(x), x) + x**4*Derivative(y(x), (x, 4)) + (a*x**4 + n*(n - 2)*(n + 1)*(n + 3))*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (-a*x**4*y(x) - n**4*y(x) - 2*n**3*y(x) + 2*n**2*x**2*Derivative(y(x), (x, 2)) + 5*n**2*y(x) + 2*n*x**2*Derivative(y(x), (x, 2)) + 6*n*y(x) - x**4*Derivative(y(x), (x, 4)))/(4*n*x*(n + 1)) cannot be solved by the factorable group method