55.32.12 problem 193

Internal problem ID [13966]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-6
Problem number : 193
Date solved : Thursday, October 02, 2025 at 08:21:56 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.023 (sec). Leaf size: 25

ode:=x^2*(a*x+b)*diff(diff(y(x),x),x)+(a*(2-n-m)*x^2-b*(m+n)*x)*diff(y(x),x)+(a*m*(n-1)*x+b*n*(1+m))*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {c_1 \,x^{n}+c_2 \,x^{m +1}}{a x +b} \]

✓ Mathematica. Time used: 0.161 (sec). Leaf size: 82

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

\begin{align*} y(x)&\to \frac {x^{\frac {1}{2} \left (-\sqrt {(m-n+1)^2}+m+n+1\right )} \left (c_2 x^{\sqrt {(m-n+1)^2}}+c_1 \sqrt {(m-n+1)^2}\right )}{\sqrt {(m-n+1)^2} (a x+b)} \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**2*(a*x + b)*Derivative(y(x), (x, 2)) + (a*x**2*(-m - n + 2) - b*x*(m + n))*Derivative(y(x), x) + (a*m*x*(n - 1) + b*n*(m + 1))*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

ValueError : Expected Expr or iterable but got None