60.5.27 problem 1564

Internal problem ID [11524]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 4, linear fourth order
Problem number : 1564
Date solved : Thursday, March 13, 2025 at 08:53:37 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}+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{align*}

✓ Maple. Time used: 0.050 (sec). Leaf size: 88

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

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

✓ Mathematica. Time used: 0.642 (sec). Leaf size: 196

ode=(3 - 12*n^2 - 4*x^4)*y[x] + (-3 + 12*n^2)*x*D[y[x],x] - (3 + 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]
 

\[ y(x)\to \frac {\left (\frac {1}{32}+\frac {i}{32}\right ) \left (i \left (c_2 x^4 \, _0F_3\left (;\frac {3}{2},2-\frac {n}{2},\frac {n}{2}+2;\frac {x^4}{64}\right )-8^{2-n} e^{-\frac {1}{2} i \pi n} x^{-2 n} \left (c_3 64^n \, _0F_3\left (;1-n,\frac {1}{2}-\frac {n}{2},-\frac {n}{2};\frac {x^4}{64}\right )+c_4 e^{i \pi n} x^{4 n} \, _0F_3\left (;\frac {n}{2}+\frac {1}{2},\frac {n}{2},n+1;\frac {x^4}{64}\right )\right )\right )+8 c_1 x^2 \, _0F_3\left (;\frac {1}{2},\frac {3}{2}-\frac {n}{2},\frac {n}{2}+\frac {3}{2};\frac {x^4}{64}\right )\right )}{x} \]

✗ 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 + 3)*Derivative(y(x), (x, 2)) + x*(12*n**2 - 3)*Derivative(y(x), x) - (12*n**2 + 4*x**4 - 3)*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)) + 12*n**2*y(x) + 4*x**4*y(x) - x**4*Derivative(y(x), (x, 4)) - 4*x**3*Derivative(y(x), (x, 3)) + 3*x**2*Derivative(y(x), (x, 2)) - 3*y(x))/(3*x*(4*n**2 - 1)) cannot be solved by the factorable group method