54.5.34 problem 1571

Internal problem ID [12830]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 4, linear fourth order
Problem number : 1571
Date solved : Friday, October 03, 2025 at 03:47:32 AM
CAS classification : [[_high_order, _with_linear_symmetries]]

\begin{align*} \nu ^{4} x^{4} y^{\prime \prime \prime \prime }+\left (4 \nu -2\right ) \nu ^{3} x^{3} y^{\prime \prime \prime }+\left (\nu -1\right ) \left (2 \nu -1\right ) \nu ^{2} x^{2} y^{\prime \prime }-\frac {b^{4} x^{\frac {2}{\nu }} y}{16}&=0 \end{align*}

✓ Maple. Time used: 0.168 (sec). Leaf size: 143

ode:=nu^4*x^4*diff(diff(diff(diff(y(x),x),x),x),x)+(4*nu-2)*nu^3*x^3*diff(diff(diff(y(x),x),x),x)+(nu-1)*(2*nu-1)*nu^2*x^2*diff(diff(y(x),x),x)-1/16*b^4*x^(2/nu)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \sqrt {x}\, \left (\operatorname {BesselY}\left (\frac {1}{{\lfloor \frac {1}{\nu }\rfloor }}, \frac {\sqrt {\frac {b^{2}}{\nu ^{2}}}\, x^{\frac {{\lfloor \frac {1}{\nu }\rfloor }}{2}}}{{\lfloor \frac {1}{\nu }\rfloor }}\right ) c_2 +\operatorname {BesselJ}\left (\frac {1}{{\lfloor \frac {1}{\nu }\rfloor }}, \frac {\sqrt {\frac {b^{2}}{\nu ^{2}}}\, x^{\frac {{\lfloor \frac {1}{\nu }\rfloor }}{2}}}{{\lfloor \frac {1}{\nu }\rfloor }}\right ) c_1 +\operatorname {BesselY}\left (\frac {1}{{\lfloor \frac {1}{\nu }\rfloor }}, \frac {\sqrt {-\frac {b^{2}}{\nu ^{2}}}\, x^{\frac {{\lfloor \frac {1}{\nu }\rfloor }}{2}}}{{\lfloor \frac {1}{\nu }\rfloor }}\right ) c_4 +\operatorname {BesselJ}\left (\frac {1}{{\lfloor \frac {1}{\nu }\rfloor }}, \frac {\sqrt {-\frac {b^{2}}{\nu ^{2}}}\, x^{\frac {{\lfloor \frac {1}{\nu }\rfloor }}{2}}}{{\lfloor \frac {1}{\nu }\rfloor }}\right ) c_3 \right ) \]

✓ Mathematica. Time used: 0.029 (sec). Leaf size: 195

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

\begin{align*} y(x)&\to 8^{-\nu -1} b^{\nu } \left (x^{2/\nu }\right )^{\nu /4} \left (4^{\nu } (4 c_1 \operatorname {Gamma}(1-\nu )-i c_2 \operatorname {Gamma}(2-\nu )) \operatorname {BesselJ}\left (-\nu ,b \sqrt [4]{x^{2/\nu }}\right )+4^{\nu } (4 c_1 \operatorname {Gamma}(1-\nu )+i c_2 \operatorname {Gamma}(2-\nu )) \operatorname {BesselI}\left (-\nu ,b \sqrt [4]{x^{2/\nu }}\right )+i^{\nu } \left ((4 c_3 \operatorname {Gamma}(\nu +1)-i c_4 \operatorname {Gamma}(\nu +2)) \operatorname {BesselJ}\left (\nu ,b \sqrt [4]{x^{2/\nu }}\right )+(4 c_3 \operatorname {Gamma}(\nu +1)+i c_4 \operatorname {Gamma}(\nu +2)) \operatorname {BesselI}\left (\nu ,b \sqrt [4]{x^{2/\nu }}\right )\right )\right ) \end{align*}

✗ Sympy

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

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