78.11.12 problem 10 (a)

Internal problem ID [18283]
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 : 10 (a)
Date solved : Tuesday, January 28, 2025 at 11:45:07 AM
CAS classification : [_Laguerre]

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

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 30

dsolve(x*diff(y(x),x$2)-(x+n)*diff(y(x),x)+n*y(x)=0,y(x), singsol=all)
 

\[ y = {\mathrm e}^{x} c_{1} +c_{2} \operatorname {WhittakerM}\left (\frac {n}{2}, \frac {n}{2}+\frac {1}{2}, x\right ) {\mathrm e}^{\frac {x}{2}} x^{\frac {n}{2}} \]

✓ Solution by Mathematica

Time used: 0.118 (sec). Leaf size: 21

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

\[ y(x)\to e^x (c_1-c_2 \Gamma (n+1,x)) \]