27.10 problem 20

Internal problem ID [10101]

Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section: Chapter 2, Second-Order Differential Equations. section 2.1.2-1 Equation of form \(y''+f(x)y'+g(x)y=0\)
Problem number: 20.
ODE order: 2.
ODE degree: 1.

CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]

Solve \begin {gather*} \boxed {y^{\prime \prime }+a x y^{\prime }+b y=0} \end {gather*}

Solution by Maple

Time used: 0.016 (sec). Leaf size: 65

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

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

Solution by Mathematica

Time used: 0.012 (sec). Leaf size: 67

DSolve[y''[x]+a*x*y'[x]+b*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to e^{-\frac {a x^2}{2}} \left (c_1 H_{\frac {b}{a}-1}\left (\frac {\sqrt {a} x}{\sqrt {2}}\right )+c_2 \, _1F_1\left (\frac {a-b}{2 a};\frac {1}{2};\frac {a x^2}{2}\right )\right ) \\ \end{align*}