Internal problem ID [10097]
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]]
\[ \boxed {y^{\prime \prime }+y^{\prime } a x +b y=0} \]
✓ Solution by Maple
Time used: 0.156 (sec). Leaf size: 65
dsolve(diff(y(x),x$2)+a*x*diff(y(x),x)+b*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} {\mathrm e}^{-\frac {a \,x^{2}}{2}} \operatorname {KummerM}\left (\frac {2 a -b}{2 a}, \frac {3}{2}, \frac {a \,x^{2}}{2}\right ) x +c_{2} {\mathrm e}^{-\frac {a \,x^{2}}{2}} \operatorname {KummerU}\left (\frac {2 a -b}{2 a}, \frac {3}{2}, \frac {a \,x^{2}}{2}\right ) x \]
✓ Solution by Mathematica
Time used: 0.011 (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 \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 ) \\ \end{align*}