55.29.13 problem 73

Internal problem ID [13846]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-3
Problem number : 73
Date solved : Friday, October 03, 2025 at 06:55:08 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.048 (sec). Leaf size: 39

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

\[ y = {\mathrm e}^{-x a} \left (\operatorname {KummerU}\left (m , m +n , \left (-b +a \right ) x \right ) c_2 +\operatorname {KummerM}\left (m , m +n , \left (-b +a \right ) x \right ) c_1 \right ) \]

✓ Mathematica. Time used: 0.056 (sec). Leaf size: 46

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

\begin{align*} y(x)&\to e^{-a x} (c_1 \operatorname {HypergeometricU}(m,m+n,(a-b) x)+c_2 L_{-m}^{m+n-1}((a-b) x)) \end{align*}

✗ Sympy

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

ValueError : Expected Expr or iterable but got None