23.5.124 problem 124
Internal
problem
ID
[6733]
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
:
124
Date
solved
:
Friday, October 03, 2025 at 02:09:51 AM
CAS
classification
:
[[_3rd_order, _with_linear_symmetries]]
\begin{align*} \left (a -x \right )^{3} \left (b -x \right )^{3} y^{\prime \prime \prime }&=c y \end{align*}
✓ Maple. Time used: 0.027 (sec). Leaf size: 437
ode:=(a-x)^3*(b-x)^3*diff(diff(diff(y(x),x),x),x) = c*y(x);
dsolve(ode,y(x), singsol=all);
\[
y = \left (x -a \right )^{-\frac {2 b}{a -b}} \left (x -b \right )^{\frac {2 a}{a -b}} \left (c_1 \left (a -x \right )^{\frac {\operatorname {RootOf}\left (\textit {\_Z}^{3}+\left (-3 a -3 b \right ) \textit {\_Z}^{2}+\left (2 a^{2}+8 a b +2 b^{2}\right ) \textit {\_Z} -4 a^{2} b -4 a \,b^{2}-c , \operatorname {index} =1\right )}{a -b}} \left (b -x \right )^{-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{3}+\left (-3 a -3 b \right ) \textit {\_Z}^{2}+\left (2 a^{2}+8 a b +2 b^{2}\right ) \textit {\_Z} -4 a^{2} b -4 a \,b^{2}-c , \operatorname {index} =1\right )}{a -b}}+c_2 \left (a -x \right )^{\frac {\operatorname {RootOf}\left (\textit {\_Z}^{3}+\left (-3 a -3 b \right ) \textit {\_Z}^{2}+\left (2 a^{2}+8 a b +2 b^{2}\right ) \textit {\_Z} -4 a^{2} b -4 a \,b^{2}-c , \operatorname {index} =2\right )}{a -b}} \left (b -x \right )^{-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{3}+\left (-3 a -3 b \right ) \textit {\_Z}^{2}+\left (2 a^{2}+8 a b +2 b^{2}\right ) \textit {\_Z} -4 a^{2} b -4 a \,b^{2}-c , \operatorname {index} =2\right )}{a -b}}+c_3 \left (a -x \right )^{\frac {\operatorname {RootOf}\left (\textit {\_Z}^{3}+\left (-3 a -3 b \right ) \textit {\_Z}^{2}+\left (2 a^{2}+8 a b +2 b^{2}\right ) \textit {\_Z} -4 a^{2} b -4 a \,b^{2}-c , \operatorname {index} =3\right )}{a -b}} \left (b -x \right )^{-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{3}+\left (-3 a -3 b \right ) \textit {\_Z}^{2}+\left (2 a^{2}+8 a b +2 b^{2}\right ) \textit {\_Z} -4 a^{2} b -4 a \,b^{2}-c , \operatorname {index} =3\right )}{a -b}}\right )
\]
✓ Mathematica. Time used: 130.071 (sec). Leaf size: 165
ode=(a - x)^3*(b - x)^3*D[y[x],{x,3}] == c*y[x];
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to c_1 (x-b)^2 \left (\frac {x-a}{x-b}\right )^{\text {Root}\left [-\text {$\#$1}^3+3 \text {$\#$1}^2-2 \text {$\#$1}+\frac {c}{(a-b)^3}\&,1\right ]}+c_2 (x-b)^2 \left (\frac {x-a}{x-b}\right )^{\text {Root}\left [-\text {$\#$1}^3+3 \text {$\#$1}^2-2 \text {$\#$1}+\frac {c}{(a-b)^3}\&,2\right ]}+c_3 (x-b)^2 \left (\frac {x-a}{x-b}\right )^{\text {Root}\left [-\text {$\#$1}^3+3 \text {$\#$1}^2-2 \text {$\#$1}+\frac {c}{(a-b)^3}\&,3\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*y(x) + (a - x)**3*(b - x)**3*Derivative(y(x), (x, 3)),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : solve: Cannot solve -c*y(x) + (a - x)**3*(b - x)**3*Derivative(y(x), (x, 3))