44.12.7 problem 6(a)

Internal problem ID [9297]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 2. Second-Order Linear Equations. Section 2.4. THE USE OF A KNOWN SOLUTION TO FIND ANOTHER. Page 74
Problem number : 6(a)
Date solved : Tuesday, September 30, 2025 at 06:16:15 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-\frac {x y^{\prime }}{x -1}+\frac {y}{x -1}&=0 \end{align*}

Using reduction of order method given that one solution is

\begin{align*} y&=x \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 12

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

\[ y = c_1 x +c_2 \,{\mathrm e}^{x} \]

✓ Mathematica. Time used: 0.091 (sec). Leaf size: 90

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

\begin{align*} y(x)&\to \exp \left (\int _1^x\frac {K[1]-2}{2 (K[1]-1)}dK[1]-\frac {1}{2} \int _1^x-\frac {K[2]}{K[2]-1}dK[2]\right ) \left (c_2 \int _1^x\exp \left (-2 \int _1^{K[3]}\frac {K[1]-2}{2 (K[1]-1)}dK[1]\right )dK[3]+c_1\right ) \end{align*}

✗ Sympy

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

False