54.4.60 problem 1518

Internal problem ID [12782]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 3, linear third order
Problem number : 1518
Date solved : Friday, October 03, 2025 at 03:47:23 AM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.010 (sec). Leaf size: 60

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

\[ y = \frac {3 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {x^{2}+1}}\right ) \sqrt {x^{2}+1}\, c_2 \,x^{2}+c_1 \,x^{2} \sqrt {x^{2}+1}+2 c_3 \,x^{3}-3 c_2 \,x^{2}+c_3 x -c_2}{x} \]

✓ Mathematica. Time used: 0.126 (sec). Leaf size: 270

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

\begin{align*} y(x)&\to \frac {1}{3} \left (2 x^2+1\right ) \left (c_2 \int _1^x\exp \left (\int _1^{K[4]}\frac {4 K[1]^4+6 K[1]^2+3}{4 K[1]^5+6 K[1]^3+2 K[1]}dK[1]-\frac {1}{2} \int _1^{K[4]}\frac {24 \left (K[2]^4+K[2]^2\right )+3}{2 K[2]^5+3 K[2]^3+K[2]}dK[2]\right )dK[4]+c_3 \int _1^x\exp \left (\int _1^{K[5]}\frac {4 K[1]^4+6 K[1]^2+3}{4 K[1]^5+6 K[1]^3+2 K[1]}dK[1]-\frac {1}{2} \int _1^{K[5]}\frac {24 \left (K[2]^4+K[2]^2\right )+3}{2 K[2]^5+3 K[2]^3+K[2]}dK[2]\right ) \int _1^{K[5]}\exp \left (-2 \int _1^{K[3]}\frac {4 K[1]^4+6 K[1]^2+3}{4 K[1]^5+6 K[1]^3+2 K[1]}dK[1]\right )dK[3]dK[5]+c_1\right ) \end{align*}

✗ Sympy

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

NotImplementedError : solve: Cannot solve x*(x**2 + 1)*Derivative(y(x), (x, 3)) + (6*x**2 + 3)*Derivative(y(x), (x, 2)) - 12*y(x)