54.4.68 problem 1526

Internal problem ID [12790]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 3, linear third order
Problem number : 1526
Date solved : Friday, October 03, 2025 at 03:47:25 AM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (x^{4}+2 x^{2}+2 x +1\right ) y^{\prime \prime \prime }-\left (2 x^{6}+3 x^{4}-6 x^{2}-6 x -1\right ) y^{\prime \prime }+\left (x^{6}-6 x^{3}-15 x^{2}-12 x -2\right ) y^{\prime }+\left (x^{4}+4 x^{3}+8 x^{2}+6 x +1\right ) y&=0 \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 19

ode:=x^2*(x^4+2*x^2+2*x+1)*diff(diff(diff(y(x),x),x),x)-(2*x^6+3*x^4-6*x^2-6*x-1)*diff(diff(y(x),x),x)+(x^6-6*x^3-15*x^2-12*x-2)*diff(y(x),x)+(x^4+4*x^3+8*x^2+6*x+1)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_2 \,{\mathrm e}^{\frac {1}{x}}+{\mathrm e}^{x} \left (c_3 x +c_1 \right ) \]

✓ Mathematica. Time used: 130.443 (sec). Leaf size: 25

ode=(1 + 6*x + 8*x^2 + 4*x^3 + x^4)*y[x] + (-2 - 12*x - 15*x^2 - 6*x^3 + x^6)*D[y[x],x] - (-1 - 6*x - 6*x^2 + 3*x^4 + 2*x^6)*D[y[x],{x,2}] + x^2*(1 + 2*x + 2*x^2 + x^4)*Derivative[3][y][x] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to e^x (c_2 x+c_1)+c_3 e^{\frac {1}{x}} \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*(x**4 + 2*x**2 + 2*x + 1)*Derivative(y(x), (x, 3)) + (x**4 + 4*x**3 + 8*x**2 + 6*x + 1)*y(x) + (x**6 - 6*x**3 - 15*x**2 - 12*x - 2)*Derivative(y(x), x) - (2*x**6 + 3*x**4 - 6*x**2 - 6*x - 1)*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (-2*x**6*Derivative(y(x), (x, 2)) + x**6*Derivative(y(x), (x, 3)) + x**4*y(x) - 3*x**4*Derivative(y(x), (x, 2)) + 2*x**4*Derivative(y(x), (x, 3)) + 4*x**3*y(x) + 2*x**3*Derivative(y(x), (x, 3)) + 8*x**2*y(x) + 6*x**2*Derivative(y(x), (x, 2)) + x**2*Derivative(y(x), (x, 3)) + 6*x*y(x) + 6*x*Derivative(y(x), (x, 2)) + y(x) + Derivative(y(x), (x, 2)))/(-x**6 + 6*x**3 + 15*x**2 + 12*x + 2) cannot be solved by the factorable group method