60.5.21 problem 1558

Internal problem ID [11518]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 4, linear fourth order
Problem number : 1558
Date solved : Sunday, March 30, 2025 at 08:24:04 PM
CAS classification : [[_high_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime \prime \prime }+\left (2 n -2 \nu +4\right ) x y^{\prime \prime \prime }+\left (n -\nu +1\right ) \left (n -\nu +2\right ) y^{\prime \prime }-\frac {b^{4} y}{16}&=0 \end{align*}

✓ Maple. Time used: 0.013 (sec). Leaf size: 67

ode:=x^2*diff(diff(diff(diff(y(x),x),x),x),x)+(2*n-2*nu+4)*x*diff(diff(diff(y(x),x),x),x)+(n-nu+1)*(n-nu+2)*diff(diff(y(x),x),x)-1/16*b^4*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \left (\operatorname {BesselK}\left (n -\nu , b \sqrt {x}\right ) c_3 +\operatorname {BesselI}\left (n -\nu , b \sqrt {x}\right ) c_1 +\operatorname {BesselY}\left (n -\nu , b \sqrt {x}\right ) c_4 +\operatorname {BesselJ}\left (n -\nu , b \sqrt {x}\right ) c_2 \right ) x^{-\frac {n}{2}+\frac {\nu }{2}} \]

✓ Mathematica. Time used: 0.09 (sec). Leaf size: 222

ode=-1/16*(b^4*y[x]) + (1 + n - nu)*(2 + n - nu)*D[y[x],{x,2}] + (4 + 2*n - 2*nu)*x*Derivative[3][y][x] + x^2*Derivative[4][y][x] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to i^{-n} 2^{n-3 \nu -3} b^{\nu -n} x^{\frac {\nu -n}{2}} \left (i^n 4^{\nu } (4 c_1 \operatorname {Gamma}(n-\nu +1)-i c_2 \operatorname {Gamma}(n-\nu +2)) \operatorname {BesselJ}\left (n-\nu ,b \sqrt {x}\right )+i^n 4^{\nu } (4 c_1 \operatorname {Gamma}(n-\nu +1)+i c_2 \operatorname {Gamma}(n-\nu +2)) \operatorname {BesselI}\left (n-\nu ,b \sqrt {x}\right )+4^n i^{\nu } \left ((4 c_3 \operatorname {Gamma}(-n+\nu +1)-i c_4 \operatorname {Gamma}(-n+\nu +2)) \operatorname {BesselJ}\left (\nu -n,b \sqrt {x}\right )+(4 c_3 \operatorname {Gamma}(-n+\nu +1)+i c_4 \operatorname {Gamma}(-n+\nu +2)) \operatorname {BesselI}\left (\nu -n,b \sqrt {x}\right )\right )\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
b = symbols("b") 
n = symbols("n") 
nu = symbols("nu") 
y = Function("y") 
ode = Eq(-b**4*y(x)/16 + x**2*Derivative(y(x), (x, 4)) + x*(2*n - 2*nu + 4)*Derivative(y(x), (x, 3)) + (n - nu + 1)*(n - nu + 2)*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : solve: Cannot solve -b**4*y(x)/16 + x**2*Derivative(y(x), (x, 4)) + x*(2*n - 2*nu + 4)*Derivative(y(x), (x, 3)) + (n - nu + 1)*(n - nu + 2)*Derivative(y(x), (x, 2))