73.9.17 problem 14.2 (g)

Internal problem ID [15278]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 14. Higher order equations and the reduction of order method. Additional exercises page 277
Problem number : 14.2 (g)
Date solved : Tuesday, January 28, 2025 at 07:51:29 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using reduction of order method given that one solution is

\begin{align*} y&={\mathrm e}^{-x} \end{align*}

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 14

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

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

✓ Solution by Mathematica

Time used: 0.205 (sec). Leaf size: 89

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

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