54.4.69 problem 1527
Internal
problem
ID
[12791]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
3,
linear
third
order
Problem
number
:
1527
Date
solved
:
Friday, October 03, 2025 at 03:47:26 AM
CAS
classification
:
[[_3rd_order, _with_linear_symmetries]]
\begin{align*} \left (x -a \right )^{3} \left (x -b \right )^{3} y^{\prime \prime \prime }-c y&=0 \end{align*}
✓ Maple. Time used: 0.115 (sec). Leaf size: 437
ode:=(x-a)^3*(x-b)^3*diff(diff(diff(y(x),x),x),x)-c*y(x) = 0;
dsolve(ode,y(x), singsol=all);
\[
y = \left (x -a \right )^{-\frac {2 b}{a -b}} \left (x -b \right )^{\frac {2 a}{a -b}} \left (c_1 \left (-x +a \right )^{\frac {\operatorname {RootOf}\left (\textit {\_Z}^{3}+\left (-3 a -3 b \right ) \textit {\_Z}^{2}+\left (2 a^{2}+8 a b +2 b^{2}\right ) \textit {\_Z} -4 a^{2} b -4 a \,b^{2}-c , \operatorname {index} =1\right )}{a -b}} \left (-x +b \right )^{-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{3}+\left (-3 a -3 b \right ) \textit {\_Z}^{2}+\left (2 a^{2}+8 a b +2 b^{2}\right ) \textit {\_Z} -4 a^{2} b -4 a \,b^{2}-c , \operatorname {index} =1\right )}{a -b}}+c_2 \left (-x +a \right )^{\frac {\operatorname {RootOf}\left (\textit {\_Z}^{3}+\left (-3 a -3 b \right ) \textit {\_Z}^{2}+\left (2 a^{2}+8 a b +2 b^{2}\right ) \textit {\_Z} -4 a^{2} b -4 a \,b^{2}-c , \operatorname {index} =2\right )}{a -b}} \left (-x +b \right )^{-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{3}+\left (-3 a -3 b \right ) \textit {\_Z}^{2}+\left (2 a^{2}+8 a b +2 b^{2}\right ) \textit {\_Z} -4 a^{2} b -4 a \,b^{2}-c , \operatorname {index} =2\right )}{a -b}}+c_3 \left (-x +a \right )^{\frac {\operatorname {RootOf}\left (\textit {\_Z}^{3}+\left (-3 a -3 b \right ) \textit {\_Z}^{2}+\left (2 a^{2}+8 a b +2 b^{2}\right ) \textit {\_Z} -4 a^{2} b -4 a \,b^{2}-c , \operatorname {index} =3\right )}{a -b}} \left (-x +b \right )^{-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{3}+\left (-3 a -3 b \right ) \textit {\_Z}^{2}+\left (2 a^{2}+8 a b +2 b^{2}\right ) \textit {\_Z} -4 a^{2} b -4 a \,b^{2}-c , \operatorname {index} =3\right )}{a -b}}\right )
\]
✓ Mathematica. Time used: 130.081 (sec). Leaf size: 165
ode=-(c*y[x]) + (-a + x)^3*(-b + x)^3*Derivative[3][y][x] == 0;
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))