23.5.181 problem 181
Internal
problem
ID
[6790]
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
:
181
Date
solved
:
Friday, October 03, 2025 at 02:09:52 AM
CAS
classification
:
[[_high_order, _with_linear_symmetries]]
\begin{align*} -b^{4} x^{\frac {2}{a}} y+16 \left (1-2 a \right ) \left (1-a \right ) a^{2} x^{2} y^{\prime \prime }-32 \left (1-2 a \right ) a^{2} x^{3} y^{\prime \prime \prime }+16 a^{4} x^{4} y^{\prime \prime \prime \prime }&=0 \end{align*}
✓ Maple. Time used: 0.015 (sec). Leaf size: 508
ode:=-b^4*x^(2/a)*y(x)+16*(1-2*a)*(-a+1)*a^2*x^2*diff(diff(y(x),x),x)-32*(1-2*a)*a^2*x^3*diff(diff(diff(y(x),x),x),x)+16*a^4*x^4*diff(diff(diff(diff(y(x),x),x),x),x) = 0;
dsolve(ode,y(x), singsol=all);
\[
y = c_1 \operatorname {hypergeom}\left (\left [\right ], \left [1-\frac {a}{2}, -\frac {5 a^{2}+\sqrt {-7 a^{4}+4 a^{3}+16 a^{2}-16 a +4}-8 a +2}{4 a}, \frac {-5 a^{2}+\sqrt {-7 a^{4}+4 a^{3}+16 a^{2}-16 a +4}+8 a -2}{4 a}\right ], \frac {b^{4} x^{\frac {2}{a}}}{256}\right )+c_2 x \operatorname {hypergeom}\left (\left [\right ], \left [1+\frac {a}{2}, -\frac {3 a^{2}+\sqrt {-7 a^{4}+4 a^{3}+16 a^{2}-16 a +4}-8 a +2}{4 a}, \frac {-3 a^{2}+\sqrt {-7 a^{4}+4 a^{3}+16 a^{2}-16 a +4}+8 a -2}{4 a}\right ], \frac {b^{4} x^{\frac {2}{a}}}{256}\right )+c_3 \,x^{\frac {5 a^{2}-4 a +2+\sqrt {-7 a^{4}+4 a^{3}+16 a^{2}-16 a +4}}{2 a^{2}}} \operatorname {hypergeom}\left (\left [\right ], \left [\frac {2 a +\sqrt {-7 a^{4}+4 a^{3}+16 a^{2}-16 a +4}}{2 a}, \frac {3 a^{2}+\sqrt {-7 a^{4}+4 a^{3}+16 a^{2}-16 a +4}+2}{4 a}, \frac {5 a^{2}+\sqrt {-7 a^{4}+4 a^{3}+16 a^{2}-16 a +4}+2}{4 a}\right ], \frac {b^{4} x^{\frac {2}{a}}}{256}\right )+c_4 \,x^{-\frac {-5 a^{2}+\sqrt {-7 a^{4}+4 a^{3}+16 a^{2}-16 a +4}+4 a -2}{2 a^{2}}} \operatorname {hypergeom}\left (\left [\right ], \left [-\frac {-2 a +\sqrt {-7 a^{4}+4 a^{3}+16 a^{2}-16 a +4}}{2 a}, -\frac {-3 a^{2}+\sqrt {-7 a^{4}+4 a^{3}+16 a^{2}-16 a +4}-2}{4 a}, -\frac {-5 a^{2}+\sqrt {-7 a^{4}+4 a^{3}+16 a^{2}-16 a +4}-2}{4 a}\right ], \frac {b^{4} x^{\frac {2}{a}}}{256}\right )
\]
✓ Mathematica. Time used: 0.034 (sec). Leaf size: 860
ode=-(b^4*x^(2/a)*y[x]) + 16*(1 - 2*a)*(1 - a)*a^2*x^2*D[y[x],{x,2}] - 32*(1 - 2*a)*a^2*x^3*D[y[x],{x,3}] + 16*a^4*x^4*D[y[x],{x,4}] == 0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to \left (\frac {i}{16}\right )^a b^{2 a} c_2 \, _0F_3\left (;\frac {a}{2}+1,-\frac {3 a}{4}+2-\frac {\sqrt {-7 a^4+4 a^3+16 a^2-16 a+4}}{4 a}-\frac {1}{2 a},-\frac {3 a}{4}+2+\frac {\sqrt {-7 a^4+4 a^3+16 a^2-16 a+4}}{4 a}-\frac {1}{2 a};\frac {1}{256} b^4 x^{2/a}\right ) \left (x^{2/a}\right )^{a/2}-(-1)^{\frac {5 a}{4}-\frac {\sqrt {-7 a^4+4 a^3+16 a^2-16 a+4}}{4 a}} i^{\frac {1}{a}} 4^{-\frac {5 a^2-4 a+\sqrt {-7 a^4+4 a^3+16 a^2-16 a+4}+2}{a}} b^{\frac {5 a^2-4 a-\sqrt {-7 a^4+4 a^3+16 a^2-16 a+4}+2}{a}} \left (i^{\frac {\sqrt {-7 a^4+4 a^3+16 a^2-16 a+4}}{a}} b^{\frac {2 \sqrt {-7 a^4+4 a^3+16 a^2-16 a+4}}{a}} c_4 \, _0F_3\left (;\frac {3 a}{4}+\frac {\sqrt {-7 a^4+4 a^3+16 a^2-16 a+4}}{4 a}+\frac {1}{2 a},\frac {5 a}{4}+\frac {\sqrt {-7 a^4+4 a^3+16 a^2-16 a+4}}{4 a}+\frac {1}{2 a},\frac {\sqrt {-7 a^4+4 a^3+16 a^2-16 a+4}}{2 a}+1;\frac {1}{256} b^4 x^{2/a}\right ) \left (x^{2/a}\right )^{\frac {\sqrt {-7 a^4+4 a^3+16 a^2-16 a+4}}{2 a}}+16^{\frac {\sqrt {-7 a^4+4 a^3+16 a^2-16 a+4}}{a}} c_3 \, _0F_3\left (;1-\frac {\sqrt {-7 a^4+4 a^3+16 a^2-16 a+4}}{2 a},\frac {3 a}{4}-\frac {\sqrt {-7 a^4+4 a^3+16 a^2-16 a+4}}{4 a}+\frac {1}{2 a},\frac {5 a}{4}-\frac {\sqrt {-7 a^4+4 a^3+16 a^2-16 a+4}}{4 a}+\frac {1}{2 a};\frac {1}{256} b^4 x^{2/a}\right )\right ) \left (x^{2/a}\right )^{-\frac {-5 a^2+4 a+\sqrt {-7 a^4+4 a^3+16 a^2-16 a+4}-2}{4 a}}+c_1 \, _0F_3\left (;1-\frac {a}{2},-\frac {5 a}{4}+2-\frac {\sqrt {-7 a^4+4 a^3+16 a^2-16 a+4}}{4 a}-\frac {1}{2 a},-\frac {5 a}{4}+2+\frac {\sqrt {-7 a^4+4 a^3+16 a^2-16 a+4}}{4 a}-\frac {1}{2 a};\frac {1}{256} b^4 x^{2/a}\right ) \end{align*}
✗ Sympy
from sympy import *
x = symbols("x")
a = symbols("a")
b = symbols("b")
y = Function("y")
ode = Eq(16*a**4*x**4*Derivative(y(x), (x, 4)) - a**2*x**3*(32 - 64*a)*Derivative(y(x), (x, 3)) + a**2*x**2*(1 - a)*(16 - 32*a)*Derivative(y(x), (x, 2)) - b**4*x**(2/a)*y(x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : solve: Cannot solve 16*a**4*x**4*Derivative(y(x), (x, 4)) - a**2*x**3*(32 - 64