50.14.28 problem 4(d)

Internal problem ID [8066]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 2. Problems for Review and Discovery. Drill excercises. Page 105
Problem number : 4(d)
Date solved : Monday, January 27, 2025 at 03:41:19 PM
CAS classification : [[_2nd_order, _missing_y]]

\begin{align*} y^{\prime \prime }+y^{\prime }&=\frac {x -1}{x} \end{align*}

Using reduction of order method given that one solution is

\begin{align*} y&=\ln \left (x \right ) \end{align*}

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 26

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

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

✓ Solution by Mathematica

Time used: 0.137 (sec). Leaf size: 30

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

\[ y(x)\to e^{-x} \operatorname {ExpIntegralEi}(x)+x-\log (x)-c_1 e^{-x}+c_2 \]