64.9.4 problem 4

Internal problem ID [13331]
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 : Monday, March 31, 2025 at 07:51:44 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 (x +1\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: 0.527 (sec). Leaf size: 119

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]
 

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

✗ 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