60.4.30 problem 1486

Internal problem ID [11453]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 3, linear third order
Problem number : 1486
Date solved : Sunday, March 30, 2025 at 08:22:32 PM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

\begin{align*} \left (2 x -1\right ) y^{\prime \prime \prime }-8 x y^{\prime }+8 y&=0 \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 51

ode:=(2*x-1)*diff(diff(diff(y(x),x),x),x)-8*x*diff(y(x),x)+8*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.167 (sec). Leaf size: 144

ode=8*y[x] - 8*x*D[y[x],x] + (-1 + 2*x)*Derivative[3][y][x] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-8*x*Derivative(y(x), x) + (2*x - 1)*Derivative(y(x), (x, 3)) + 8*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (x*Derivative(y(x), (x, 3))/4 + y(x) - Derivative(y(x), (x, 3))/8)/x cannot be solved by the factorable group method