64.9.4 problem 4

Internal problem ID [13402]
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 : Tuesday, January 28, 2025 at 05:41:48 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*}

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 15

dsolve([(x^2-x+1)*diff(y(x),x$2)-(x^2+x)*diff(y(x),x)+(x+1)*y(x)=0,x],singsol=all)
 

\[ y = c_{1} x +c_{2} \left (x -1\right ) {\mathrm e}^{x} \]

✓ Solution by Mathematica

Time used: 0.527 (sec). Leaf size: 119

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

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