60.5.28 problem 1565

Internal problem ID [11525]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 4, linear fourth order
Problem number : 1565
Date solved : Sunday, March 30, 2025 at 08:24:11 PM
CAS classification : [[_high_order, _with_linear_symmetries]]

\begin{align*} x^{4} y^{\prime \prime \prime \prime }+6 x^{3} y^{\prime \prime \prime }+\left (4 x^{4}+\left (-\rho ^{2}-\sigma ^{2}+7\right ) x^{2}\right ) y^{\prime \prime }+\left (16 x^{3}+\left (-\rho ^{2}-\sigma ^{2}+1\right ) x \right ) y^{\prime }+\left (\rho ^{2} \sigma ^{2}+8 x^{2}\right ) y&=0 \end{align*}

✓ Maple. Time used: 0.013 (sec). Leaf size: 71

ode:=x^4*diff(diff(diff(diff(y(x),x),x),x),x)+6*x^3*diff(diff(diff(y(x),x),x),x)+(4*x^4+(-rho^2-sigma^2+7)*x^2)*diff(diff(y(x),x),x)+(16*x^3+(-rho^2-sigma^2+1)*x)*diff(y(x),x)+(rho^2*sigma^2+8*x^2)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \left (\operatorname {BesselY}\left (-\frac {\sigma }{2}+\frac {\rho }{2}, x\right ) c_2 +c_1 \operatorname {BesselJ}\left (-\frac {\sigma }{2}+\frac {\rho }{2}, x\right )\right ) \operatorname {BesselJ}\left (\frac {\sigma }{2}+\frac {\rho }{2}, x\right )+\operatorname {BesselY}\left (\frac {\sigma }{2}+\frac {\rho }{2}, x\right ) \left (\operatorname {BesselY}\left (-\frac {\sigma }{2}+\frac {\rho }{2}, x\right ) c_4 +c_3 \operatorname {BesselJ}\left (-\frac {\sigma }{2}+\frac {\rho }{2}, x\right )\right ) \]

✓ Mathematica. Time used: 0.264 (sec). Leaf size: 242

ode=(rho^2*sigma^2 + 8*x^2)*y[x] + ((1 - rho^2 - sigma^2)*x + 16*x^3)*D[y[x],x] + ((7 - rho^2 - sigma^2)*x^2 + 4*x^4)*D[y[x],{x,2}] + 6*x^3*Derivative[3][y][x] + x^4*Derivative[4][y][x] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to c_1 x^{-\rho } \, _2F_3\left (\frac {1}{2}-\frac {\rho }{2},1-\frac {\rho }{2};1-\rho ,-\frac {\rho }{2}-\frac {\sigma }{2}+1,-\frac {\rho }{2}+\frac {\sigma }{2}+1;-x^2\right )+c_3 x^{-\sigma } \, _2F_3\left (\frac {1}{2}-\frac {\sigma }{2},1-\frac {\sigma }{2};1-\sigma ,-\frac {\rho }{2}-\frac {\sigma }{2}+1,\frac {\rho }{2}-\frac {\sigma }{2}+1;-x^2\right )+c_4 x^{\sigma } \, _2F_3\left (\frac {\sigma }{2}+\frac {1}{2},\frac {\sigma }{2}+1;-\frac {\rho }{2}+\frac {\sigma }{2}+1,\frac {\rho }{2}+\frac {\sigma }{2}+1,\sigma +1;-x^2\right )+c_2 x^{\rho } \, _2F_3\left (\frac {\rho }{2}+\frac {1}{2},\frac {\rho }{2}+1;\rho +1,\frac {\rho }{2}-\frac {\sigma }{2}+1,\frac {\rho }{2}+\frac {\sigma }{2}+1;-x^2\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
rho = symbols("rho") 
sigma = symbols("sigma") 
y = Function("y") 
ode = Eq(x**4*Derivative(y(x), (x, 4)) + 6*x**3*Derivative(y(x), (x, 3)) + (16*x**3 + x*(-rho**2 - sigma**2 + 1))*Derivative(y(x), x) + (4*x**4 + x**2*(-rho**2 - sigma**2 + 7))*Derivative(y(x), (x, 2)) + (rho**2*sigma**2 + 8*x**2)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (rho**2*sigma**2*y(x) - rho**2*x**2*Derivative(y(x), (x, 2)) - sigma**2*x**2*Derivative(y(x), (x, 2)) + 4*x**4*Derivative(y(x), (x, 2)) + x**4*Derivative(y(x), (x, 4)) + 6*x**3*Derivative(y(x), (x, 3)) + 8*x**2*y(x) + 7*x**2*Derivative(y(x), (x, 2)))/(x*(rho**2 + sigma**2 - 16*x**2 - 1)) cannot be solved by the factorable group method