67.2.56 problem Problem 20(c)

Internal problem ID [13942]
Book : APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A. Dobrushkin. CRC Press 2015
Section : Chapter 4, Second and Higher Order Linear Differential Equations. Problems page 221
Problem number : Problem 20(c)
Date solved : Monday, March 31, 2025 at 08:19:44 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 26

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

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

✓ Mathematica. Time used: 0.421 (sec). Leaf size: 77

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

\[ y(x)\to \frac {\exp \left (\int _1^x\left (\frac {1}{2 K[1]+2}-\frac {1}{K[1]}\right )dK[1]\right ) \left (c_2 \int _1^x\exp \left (-2 \int _1^{K[2]}\left (\frac {1}{2 K[1]+2}-\frac {1}{K[1]}\right )dK[1]\right )dK[2]+c_1\right )}{\sqrt {x+1}} \]

✗ Sympy

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

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