73.9.20 problem 14.2 (j)

Internal problem ID [15281]
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 (j)
Date solved : Tuesday, January 28, 2025 at 07:51:31 AM
CAS classification : [[_2nd_order, _exact, _linear, _homogeneous]]

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

Using reduction of order method given that one solution is

\begin{align*} y&=\frac {1}{x} \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 16

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

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

✓ Solution by Mathematica

Time used: 0.028 (sec). Leaf size: 24

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

\[ y(x)\to \frac {2 c_1 e^{-2 x}+c_2}{2 x} \]