23.5.170 problem 170

Internal problem ID [6779]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 5. THE EQUATION IS LINEAR AND OF ORDER GREATER THAN TWO, page 410
Problem number : 170
Date solved : Friday, October 03, 2025 at 02:09:51 AM
CAS classification : [[_high_order, _with_linear_symmetries]]

\begin{align*} -a^{2} y+12 y^{\prime \prime }+8 x y^{\prime \prime \prime }+x^{2} y^{\prime \prime \prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.009 (sec). Leaf size: 159

ode:=-a^2*y(x)+12*diff(diff(y(x),x),x)+8*x*diff(diff(diff(y(x),x),x),x)+x^2*diff(diff(diff(diff(y(x),x),x),x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = -\frac {-\operatorname {BesselJ}\left (1, 2 \sqrt {-a}\, \sqrt {x}\right ) c_3 \sqrt {a}-\operatorname {BesselY}\left (1, 2 \sqrt {-a}\, \sqrt {x}\right ) c_4 \sqrt {a}+\sqrt {-a}\, \left (\operatorname {BesselJ}\left (0, 2 \sqrt {-a}\, \sqrt {x}\right ) \sqrt {a}\, \sqrt {x}\, c_3 +\operatorname {BesselY}\left (0, 2 \sqrt {-a}\, \sqrt {x}\right ) \sqrt {a}\, \sqrt {x}\, c_4 -\operatorname {BesselJ}\left (1, 2 \sqrt {a}\, \sqrt {x}\right ) c_1 -\operatorname {BesselY}\left (1, 2 \sqrt {a}\, \sqrt {x}\right ) c_2 +\sqrt {x}\, \sqrt {a}\, \left (\operatorname {BesselJ}\left (0, 2 \sqrt {a}\, \sqrt {x}\right ) c_1 +\operatorname {BesselY}\left (0, 2 \sqrt {a}\, \sqrt {x}\right ) c_2 \right )\right )}{\sqrt {-a}\, x^{{3}/{2}} \sqrt {a}} \]

✓ Mathematica. Time used: 0.02 (sec). Leaf size: 146

ode=-(a^2*y[x]) + 12*D[y[x],{x,2}] + 8*x*D[y[x],{x,3}] + x^2*D[y[x],{x,4}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to c_4 G_{0,4}^{2,0}\left (\frac {a^2 x^2}{16}| \begin {array}{c} -1,0,-\frac {1}{2},\frac {1}{2} \\ \end {array} \right )+c_2 G_{0,4}^{2,0}\left (\frac {a^2 x^2}{16}| \begin {array}{c} -\frac {1}{2},\frac {1}{2},-1,0 \\ \end {array} \right )-\frac {3 i c_1 \left (\operatorname {BesselI}\left (2,2 \sqrt {a} \sqrt {x}\right )-\operatorname {BesselJ}\left (2,2 \sqrt {a} \sqrt {x}\right )\right )}{4 a x}-\frac {c_3 \left (\operatorname {BesselJ}\left (2,2 \sqrt {a} \sqrt {x}\right )+\operatorname {BesselI}\left (2,2 \sqrt {a} \sqrt {x}\right )\right )}{a x} \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(-a**2*y(x) + x**2*Derivative(y(x), (x, 4)) + 8*x*Derivative(y(x), (x, 3)) + 12*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : solve: Cannot solve -a**2*y(x) + x**2*Derivative(y(x), (x, 4)) + 8*x*Derivativ