23.5.173 problem 173

Internal problem ID [6782]
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 : 173
Date solved : Friday, October 03, 2025 at 02:09:52 AM
CAS classification : [[_high_order, _with_linear_symmetries]]

\begin{align*} -a^{4} x^{3} y-x y^{\prime \prime }+2 x^{2} y^{\prime \prime \prime }+x^{3} y^{\prime \prime \prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.010 (sec). Leaf size: 148

ode:=-a^4*x^3*y(x)-x*diff(diff(y(x),x),x)+2*x^2*diff(diff(diff(y(x),x),x),x)+x^3*diff(diff(diff(diff(y(x),x),x),x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_1 \operatorname {hypergeom}\left (\left [\right ], \left [\frac {3}{4}, \frac {5}{8}+\frac {\sqrt {5}}{8}, \frac {5}{8}-\frac {\sqrt {5}}{8}\right ], \frac {a^{4} x^{4}}{256}\right )+c_2 x \operatorname {hypergeom}\left (\left [\right ], \left [\frac {5}{4}, \frac {7}{8}+\frac {\sqrt {5}}{8}, \frac {7}{8}-\frac {\sqrt {5}}{8}\right ], \frac {a^{4} x^{4}}{256}\right )+c_3 \,x^{\frac {3}{2}-\frac {\sqrt {5}}{2}} \operatorname {hypergeom}\left (\left [\right ], \left [1-\frac {\sqrt {5}}{4}, \frac {9}{8}-\frac {\sqrt {5}}{8}, \frac {11}{8}-\frac {\sqrt {5}}{8}\right ], \frac {a^{4} x^{4}}{256}\right )+c_4 \,x^{\frac {3}{2}+\frac {\sqrt {5}}{2}} \operatorname {hypergeom}\left (\left [\right ], \left [1+\frac {\sqrt {5}}{4}, \frac {9}{8}+\frac {\sqrt {5}}{8}, \frac {11}{8}+\frac {\sqrt {5}}{8}\right ], \frac {a^{4} x^{4}}{256}\right ) \]

✓ Mathematica. Time used: 0.226 (sec). Leaf size: 305

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

\begin{align*} y(x)&\to c_1 \, _0F_3\left (;\frac {3}{4},\frac {5}{8}-\frac {\sqrt {5}}{8},\frac {5}{8}+\frac {\sqrt {5}}{8};\frac {a^4 x^4}{256}\right )+\frac {1}{8} \left ((1+i) \sqrt {2} a c_2 x \, _0F_3\left (;\frac {5}{4},\frac {7}{8}-\frac {\sqrt {5}}{8},\frac {7}{8}+\frac {\sqrt {5}}{8};\frac {a^4 x^4}{256}\right )+(-1)^{\frac {1}{8} \left (3-\sqrt {5}\right )} 2^{-\sqrt {5}} a^{\frac {3}{2}-\frac {\sqrt {5}}{2}} x^{\frac {3}{2}-\frac {\sqrt {5}}{2}} \left (4^{\sqrt {5}} c_3 \, _0F_3\left (;1-\frac {\sqrt {5}}{4},\frac {9}{8}-\frac {\sqrt {5}}{8},\frac {11}{8}-\frac {\sqrt {5}}{8};\frac {a^4 x^4}{256}\right )+(-1)^{\frac {\sqrt {5}}{4}} a^{\sqrt {5}} c_4 x^{\sqrt {5}} \, _0F_3\left (;\frac {9}{8}+\frac {\sqrt {5}}{8},\frac {11}{8}+\frac {\sqrt {5}}{8},1+\frac {\sqrt {5}}{4};\frac {a^4 x^4}{256}\right )\right )\right ) \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE a**4*x**2*y(x) - x**2*Derivative(y(x), (x, 4)) - 2*x*Derivative(