82.48.7 problem Ex. 7

Internal problem ID [18917]
Book : Introductory Course On Differential Equations by Daniel A Murray. Longmans Green and Co. NY. 1924
Section : Chapter VIII. End of chapter problems at page 107
Problem number : Ex. 7
Date solved : Monday, March 31, 2025 at 06:24:52 PM
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&=\frac {2}{x^{3}} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 58

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) = 2/x^3; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.117 (sec). Leaf size: 63

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]==2/x^3; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {-\arctan \left (\frac {1}{\sqrt {x^2-1}}\right )-c_3 \text {arctanh}\left (\frac {x}{\sqrt {x^2-1}}\right )+c_1 \sqrt {x^2-1}-c_2}{x \sqrt {x^2-1}} \]

✗ 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) - 2/x**3,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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