60.4.48 problem 1504

Internal problem ID [11471]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 3, linear third order
Problem number : 1504
Date solved : Sunday, March 30, 2025 at 08:22:55 PM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.008 (sec). Leaf size: 18

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

\[ y = c_1 \,x^{2}+c_2 \cos \left (x \right )+c_3 \sin \left (x \right ) \]

✓ Mathematica. Time used: 0.199 (sec). Leaf size: 212

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

\[ y(x)\to \frac {1}{2} x^2 \left (c_2 \int _1^x\exp \left (\int _1^{K[4]}\left (-\frac {K[1]}{K[1]^2+2}-i\right )dK[1]-\frac {1}{2} \int _1^{K[4]}\frac {4 K[2]^2+12}{K[2]^3+2 K[2]}dK[2]\right ) (K[4]-2 i)dK[4]+c_3 \int _1^x\exp \left (\int _1^{K[5]}\left (-\frac {K[1]}{K[1]^2+2}-i\right )dK[1]-\frac {1}{2} \int _1^{K[5]}\frac {4 K[2]^2+12}{K[2]^3+2 K[2]}dK[2]\right ) (K[5]-2 i) \int _1^{K[5]}\frac {\exp \left (-2 \int _1^{K[3]}\left (-\frac {K[1]}{K[1]^2+2}-i\right )dK[1]\right )}{(K[3]-2 i)^2}dK[3]dK[5]+c_1\right ) \]

✗ Sympy

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

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