61.27.12 problem 22

Internal problem ID [12443]
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 : Monday, March 31, 2025 at 05:34:36 AM
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*}

✓ Maple. Time used: 0.010 (sec). Leaf size: 82

ode:=diff(diff(y(x),x),x)+a*x*diff(y(x),x)+(b*x+c)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = {\mathrm e}^{-\frac {b x}{a}} \left (\operatorname {KummerU}\left (\frac {c \,a^{2}+b^{2}}{2 a^{3}}, \frac {1}{2}, -\frac {\left (x \,a^{2}-2 b \right )^{2}}{2 a^{3}}\right ) c_2 +\operatorname {KummerM}\left (\frac {c \,a^{2}+b^{2}}{2 a^{3}}, \frac {1}{2}, -\frac {\left (x \,a^{2}-2 b \right )^{2}}{2 a^{3}}\right ) c_1 \right ) \]

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

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

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

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
c = symbols("c") 
y = Function("y") 
ode = Eq(a*x*Derivative(y(x), x) + (b*x + c)*y(x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False