47.5.4 problem 4

Internal problem ID [7493]
Book : Ordinary differential equations and calculus of variations. Makarets and Reshetnyak. Wold Scientific. Singapore. 1995
Section : Chapter 2. Linear homogeneous equations. Section 2.3.4 problems. page 104
Problem number : 4
Date solved : Sunday, March 30, 2025 at 12:10:37 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 26

ode:=x^3*diff(diff(y(x),x),x)+x*diff(y(x),x)-y(x) = cos(1/x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = -\frac {x \left (-2 c_2 \,{\mathrm e}^{\frac {1}{x}}+\sin \left (\frac {1}{x}\right )+\cos \left (\frac {1}{x}\right )-2 c_1 \right )}{2} \]

✓ Mathematica. Time used: 0.118 (sec). Leaf size: 32

ode=x^3*D[y[x],{x,2}]+x*D[y[x],x]-y[x]==Cos[1/x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to -\frac {1}{2} x \left (\sin \left (\frac {1}{x}\right )+\cos \left (\frac {1}{x}\right )-2 \left (c_1 e^{\frac {1}{x}}+c_2\right )\right ) \]

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) - (-x**3*Derivative(y(x), (x, 2)) + y(x) + cos(1/x))/x cannot be solved by the factorable group method