60.4.29 problem 1485

Internal problem ID [11452]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 3, linear third order
Problem number : 1485
Date solved : Sunday, March 30, 2025 at 08:22:30 PM
CAS classification : [[_3rd_order, _exact, _linear, _homogeneous]]

\begin{align*} \left (x -2\right ) x y^{\prime \prime \prime }-\left (x -2\right ) x y^{\prime \prime }-2 y^{\prime }+2 y&=0 \end{align*}

✓ Maple. Time used: 0.010 (sec). Leaf size: 51

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

\[ y = c_3 \,\operatorname {Ei}_{1}\left (x -2\right ) {\mathrm e}^{x -2}+\frac {c_3 \,x^{2} \ln \left (x -2\right )}{4}+c_2 \,{\mathrm e}^{x}-\frac {c_3 \,x^{2} \ln \left (x \right )}{4}+\frac {\left (2 x +2\right ) c_3}{4}+c_1 \,x^{2} \]

✓ Mathematica. Time used: 0.092 (sec). Leaf size: 126

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

\[ y(x)\to x^2 \left (c_2 \int _1^x\frac {\exp \left (\frac {K[3]}{2}+\int _1^{K[3]}\left (\frac {1}{2}+\frac {1}{K[1]-2}\right )dK[1]\right )}{K[3]^3}dK[3]+c_3 \int _1^x\frac {\exp \left (\frac {K[4]}{2}+\int _1^{K[4]}\left (\frac {1}{2}+\frac {1}{K[1]-2}\right )dK[1]\right ) \int _1^{K[4]}\exp \left (-2 \int _1^{K[2]}\left (\frac {1}{2}+\frac {1}{K[1]-2}\right )dK[1]\right )dK[2]}{K[4]^3}dK[4]+c_1\right ) \]

✗ Sympy

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

NotImplementedError : The given ODE x**2*Derivative(y(x), (x, 2))/2 - x**2*Derivative(y(x), (x, 3))/2 - x*Derivative(y(x), (x, 2)) + x*Derivative(y(x), (x, 3)) - y(x) + Derivative(y(x), x) cannot be solved by the factorable group method