61.27.12 problem 22

Internal problem ID [12522]
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 : 22
Date solved : Tuesday, January 28, 2025 at 08:02:05 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.092 (sec). Leaf size: 82

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

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

✓ Solution by Mathematica

Time used: 0.052 (sec). Leaf size: 108

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

\[ y(x)\to e^{\frac {b x}{a}-\frac {a x^2}{2}} \left (c_2 \operatorname {Hypergeometric1F1}\left (-\frac {-a^3+c a^2+b^2}{2 a^3},\frac {1}{2},\frac {\left (a^2 x-2 b\right )^2}{2 a^3}\right )+c_1 \operatorname {HermiteH}\left (\frac {b^2}{a^3}+\frac {c}{a}-1,\frac {a^2 x-2 b}{\sqrt {2} a^{3/2}}\right )\right ) \]