78.11.7 problem 7 (a)

Internal problem ID [18278]
Book : DIFFERENTIAL EQUATIONS WITH APPLICATIONS AND HISTORICAL NOTES by George F. Simmons. 3rd edition. 2017. CRC press, Boca Raton FL.
Section : Chapter 3. Second order linear equations. Section 16. The Use of a Known Solution to find Another. Problems at page 121
Problem number : 7 (a)
Date solved : Tuesday, January 28, 2025 at 11:45:04 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-\frac {x y^{\prime }}{x -1}+\frac {y}{x -1}&=0 \end{align*}

Using reduction of order method given that one solution is

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

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 12

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

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

✓ Solution by Mathematica

Time used: 0.025 (sec). Leaf size: 17

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

\[ y(x)\to c_1 e^x-c_2 x \]