23.5.172 problem 172

Internal problem ID [6781]
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 : 172
Date solved : Friday, October 03, 2025 at 02:09:51 AM
CAS classification : [[_high_order, _with_linear_symmetries]]

\begin{align*} -c^{4} y+16 \left (1+a -b \right ) \left (2+a -b \right ) y^{\prime \prime }+32 \left (2+a -b \right ) x y^{\prime \prime \prime }+16 x^{2} y^{\prime \prime \prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.012 (sec). Leaf size: 67

ode:=-c^4*y(x)+16*(1+a-b)*(2+a-b)*diff(diff(y(x),x),x)+32*(2+a-b)*x*diff(diff(diff(y(x),x),x),x)+16*x^2*diff(diff(diff(diff(y(x),x),x),x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.047 (sec). Leaf size: 222

ode=-(c^4*y[x]) + 16*(1 + a - b)*(2 + a - b)*D[y[x],{x,2}] + 32*(2 + a - b)*x*D[y[x],{x,3}] + 16*x^2*D[y[x],{x,4}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
c = symbols("c") 
y = Function("y") 
ode = Eq(-c**4*y(x) + 16*x**2*Derivative(y(x), (x, 4)) + x*(32*a - 32*b + 64)*Derivative(y(x), (x, 3)) + (a - b + 2)*(16*a - 16*b + 16)*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : solve: Cannot solve -c**4*y(x) + 16*x**2*Derivative(y(x), (x, 4)) + x*(32*a -