76.12.25 problem 37

Internal problem ID [17581]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 4. Second order linear equations. Section 4.2 (Theory of second order linear homogeneous equations). Problems at page 226
Problem number : 37
Date solved : Tuesday, January 28, 2025 at 10:44:29 AM
CAS classification : [_Laguerre]

\begin{align*} x y^{\prime \prime }-\left (x +n \right ) y^{\prime }+n 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: 0.006 (sec). Leaf size: 30

dsolve([x*diff(y(x),x$2)-(x+n)*diff(y(x),x)+n*y(x)=0,exp(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: 0.120 (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]
 

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