61.27.10 problem 20

Internal problem ID [12520]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-2
Problem number : 20
Date solved : Tuesday, January 28, 2025 at 08:02:03 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+a x y^{\prime }+b y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.292 (sec). Leaf size: 58

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

\[ y = {\mathrm e}^{-\frac {a \,x^{2}}{2}} x \left (\operatorname {KummerM}\left (\frac {2 a -b}{2 a}, \frac {3}{2}, \frac {a \,x^{2}}{2}\right ) c_{1} +\operatorname {KummerU}\left (\frac {2 a -b}{2 a}, \frac {3}{2}, \frac {a \,x^{2}}{2}\right ) c_{2} \right ) \]

✓ Solution by Mathematica

Time used: 0.044 (sec). Leaf size: 67

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

\[ y(x)\to e^{-\frac {a x^2}{2}} \left (c_1 \operatorname {HermiteH}\left (\frac {b}{a}-1,\frac {\sqrt {a} x}{\sqrt {2}}\right )+c_2 \operatorname {Hypergeometric1F1}\left (\frac {a-b}{2 a},\frac {1}{2},\frac {a x^2}{2}\right )\right ) \]