47.5.5 problem 5

Internal problem ID [7494]
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 : 5
Date solved : Sunday, March 30, 2025 at 12:10:39 PM
CAS classification : [[_2nd_order, _exact, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 42

ode:=x*(1+x)*diff(diff(y(x),x),x)+(x+2)*diff(y(x),x)-y(x) = x+1/x; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {2 \ln \left (x \right ) x^{2}+4 c_2 \,x^{2}+4 \ln \left (x \right ) x +8 c_2 x +4 c_1 +4 c_2 +6 x +5}{4 x} \]

✓ Mathematica. Time used: 0.047 (sec). Leaf size: 37

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

\[ y(x)\to \frac {1}{2} (x+2) \log (x)+\frac {1+c_1}{x}+\frac {1}{4} (-1+2 c_2) x+1+c_2 \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*(x + 1)*Derivative(y(x), (x, 2)) - x + (x + 2)*Derivative(y(x), x) - y(x) - 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)) - x**2*Derivative(y(x), (x, 2)) + x**2 + x*y(x) + 1)/(x*(x + 2)) cannot be solved by the factorable group method