55.34.13 problem 251

Internal problem ID [14024]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-8. Other equations.
Problem number : 251
Date solved : Friday, October 03, 2025 at 07:23:30 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.121 (sec). Leaf size: 65

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

\[ y = \left (x^{n +1-a} c_2 \operatorname {hypergeom}\left (\left [\frac {b +n}{n}, \frac {n +1-a}{n}\right ], \left [\frac {2 n +1-a}{n}\right ], -x^{n}\right )+c_1 \right ) \left (x^{n}+1\right )^{\frac {b}{n}} \]

✓ Mathematica. Time used: 0.058 (sec). Leaf size: 69

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

\begin{align*} y(x)&\to c_2 \left (x^n\right )^{\frac {-a+n+1}{n}} \operatorname {Hypergeometric2F1}\left (1,\frac {-a-b+n+1}{n},\frac {-a+2 n+1}{n},-x^n\right )+c_1 \left (x^n+1\right )^{b/n} \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
n = symbols("n") 
y = Function("y") 
ode = Eq(b*x**(n - 1)*(1 - a)*y(x) + x*(x**n + 1)*Derivative(y(x), (x, 2)) + (a - n + x**n*(a - b))*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

TypeError : Add object cannot be interpreted as an integer