58.9.4 problem 4

Internal problem ID [14683]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 4, Section 4.1. Basic theory of linear differential equations. Exercises page 124
Problem number : 4
Date solved : Thursday, October 02, 2025 at 09:50:11 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using reduction of order method given that one solution is

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 15

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

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

✓ Mathematica. Time used: 0.079 (sec). Leaf size: 19

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

\begin{align*} y(x)&\to c_1 x+c_2 e^x (x-1) \end{align*}

✗ Sympy

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

False