61.28.13 problem 73

Internal problem ID [12573]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-3
Problem number : 73
Date solved : Tuesday, January 28, 2025 at 08:02:26 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.152 (sec). Leaf size: 39

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

\[ y = {\mathrm e}^{-a x} \left (\operatorname {KummerU}\left (m , m +n , x \left (a -b \right )\right ) c_{2} +\operatorname {KummerM}\left (m , m +n , x \left (a -b \right )\right ) c_{1} \right ) \]

✓ Solution by Mathematica

Time used: 0.084 (sec). Leaf size: 46

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

\[ y(x)\to e^{-a x} (c_1 \operatorname {HypergeometricU}(m,m+n,(a-b) x)+c_2 L_{-m}^{m+n-1}((a-b) x)) \]