56.36.4 problem Ex 4

Internal problem ID [14283]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter IX, Miscellaneous methods for solving equations of higher order than first. Article 60. Exact equation. Integrating factor. Page 139
Problem number : Ex 4
Date solved : Thursday, October 02, 2025 at 09:27:43 AM
CAS classification : [[_3rd_order, _fully, _exact, _linear]]

\begin{align*} \left (x^{3}-x \right ) y^{\prime \prime \prime }+\left (8 x^{2}-3\right ) y^{\prime \prime }+14 x y^{\prime }+4 y&=0 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 41

ode:=(x^3-x)*diff(diff(diff(y(x),x),x),x)+(8*x^2-3)*diff(diff(y(x),x),x)+14*x*diff(y(x),x)+4*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.05 (sec). Leaf size: 50

ode=(x^3-x)*D[y[x],{x,3}]+(8*x^2-3)*D[y[x],{x,2}]+14*x*D[y[x],x]+4*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {-\frac {c_3 \text {arctanh}\left (\frac {x}{\sqrt {x^2-1}}\right )}{\sqrt {x^2-1}}-\frac {c_2}{\sqrt {x^2-1}}+c_1}{x} \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(14*x*Derivative(y(x), x) + (8*x**2 - 3)*Derivative(y(x), (x, 2)) + (x**3 - x)*Derivative(y(x), (x, 3)) + 4*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (x*(-x**2*Derivative(y(x), (x, 3)) - 8*x*Derivative(y(x), (x, 2)) + Derivative(y(x), (x, 3))) - 4*y(x) + 3*Derivative(y(x), (x, 2)))/(14*x) cannot be solved by the factorable group method