60.5.26 problem 1563

Internal problem ID [11523]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 4, linear fourth order
Problem number : 1563
Date solved : Sunday, March 30, 2025 at 08:24:09 PM
CAS classification : [[_high_order, _with_linear_symmetries]]

\begin{align*} x^{4} y^{\prime \prime \prime \prime }+4 x^{3} y^{\prime \prime \prime }-\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{align*}

✓ Maple. Time used: 0.016 (sec). Leaf size: 87

ode:=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; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {c_4 \operatorname {hypergeom}\left (\left [\right ], \left [\frac {1}{2}, \frac {1}{2}-\frac {n}{2}, \frac {1}{2}+\frac {n}{2}\right ], \frac {x^{4}}{64}\right )+\left (\operatorname {hypergeom}\left (\left [\right ], \left [\frac {3}{2}, 1-\frac {n}{2}, 1+\frac {n}{2}\right ], \frac {x^{4}}{64}\right ) c_3 +c_2 \operatorname {KelvinBei}\left (-n , x\right )^{2}+\operatorname {KelvinBer}\left (-n , x\right )^{2} c_2 +c_1 \left (\operatorname {KelvinBer}\left (n , x\right )^{2}+\operatorname {KelvinBei}\left (n , x\right )^{2}\right )\right ) x^{2}}{x} \]

✓ Mathematica. Time used: 0.944 (sec). Leaf size: 187

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

\[ y(x)\to \frac {\sqrt [4]{-1} \left (x^2 \left (c_2 \, _0F_3\left (;\frac {3}{2},1-\frac {n}{2},\frac {n}{2}+1;\frac {x^4}{64}\right )+c_3 \left (\frac {i}{8}\right )^{-n} x^{-2 n} \, _0F_3\left (;1-n,1-\frac {n}{2},\frac {3}{2}-\frac {n}{2};\frac {x^4}{64}\right )+c_4 \left (\frac {i}{8}\right )^n x^{2 n} \, _0F_3\left (;\frac {n}{2}+1,\frac {n}{2}+\frac {3}{2},n+1;\frac {x^4}{64}\right )\right )-8 i c_1 \, _0F_3\left (;\frac {1}{2},\frac {1}{2}-\frac {n}{2},\frac {n}{2}+\frac {1}{2};\frac {x^4}{64}\right )\right )}{2 \sqrt {2} x} \]

✗ Sympy

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

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