55.29.34 problem 94

Internal problem ID [13867]
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 : 94
Date solved : Thursday, October 02, 2025 at 08:08:03 AM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

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

✓ Maple. Time used: 0.113 (sec). Leaf size: 53

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

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

✓ Mathematica. Time used: 0.07 (sec). Leaf size: 121

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

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

✗ Sympy

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

ValueError : Expected Expr or iterable but got None