61.27.19 problem 29

Internal problem ID [12529]
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 : 29
Date solved : Tuesday, January 28, 2025 at 08:02:08 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+c \left (\left (a -c \right ) x^{2}+b x +1\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 36

dsolve(diff(y(x),x$2)+(a*x+b)*diff(y(x),x)+c*((a-c)*x^2+b*x+1)*y(x)=0,y(x), singsol=all)
 

\[ y = {\mathrm e}^{-\frac {c \,x^{2}}{2}} \left (c_{1} +\operatorname {erf}\left (\frac {\left (a -2 c \right ) x +b}{\sqrt {2 a -4 c}}\right ) c_{2} \right ) \]

✓ Solution by Mathematica

Time used: 0.082 (sec). Leaf size: 81

DSolve[D[y[x],{x,2}]+(a*x+b)*D[y[x],x]+c*((a-c)*x^2+b*x+1)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

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