55.28.18 problem 28

Internal problem ID [13801]
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 : 28
Date solved : Friday, October 03, 2025 at 06:54:51 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.011 (sec). Leaf size: 98

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

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

✓ Mathematica. Time used: 0.045 (sec). Leaf size: 132

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

\begin{align*} y(x)&\to e^{\frac {c x}{a}-\frac {a x^2}{2}-b x} \left (c_2 \operatorname {Hypergeometric1F1}\left (\frac {a^3-d a^2+b c a-c^2}{2 a^3},\frac {1}{2},\frac {\left (x a^2+b a-2 c\right )^2}{2 a^3}\right )+c_1 \operatorname {HermiteH}\left (\frac {-a^3+d a^2-b c a+c^2}{a^3},\frac {x a^2+b a-2 c}{\sqrt {2} a^{3/2}}\right )\right ) \end{align*}

✗ Sympy

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

False