55.29.32 problem 92

Internal problem ID [13865]
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 : 92
Date solved : Thursday, October 02, 2025 at 08:08:01 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.146 (sec). Leaf size: 92

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

\[ y = \frac {c_1}{x}+c_2 \,{\mathrm e}^{-\frac {a \,x^{n}}{2 n}} x^{-\frac {3 n}{2}-\frac {1}{2}} \left (n \left (a \,x^{n}+n +1\right ) \operatorname {WhittakerM}\left (-\frac {1}{2}+\frac {1}{2 n}, \frac {1}{2 n}+1, \frac {a \,x^{n}}{n}\right )+\operatorname {WhittakerM}\left (\frac {1}{2 n}+\frac {1}{2}, \frac {1}{2 n}+1, \frac {a \,x^{n}}{n}\right ) \left (1+n \right )^{2}\right ) \]

✓ Mathematica. Time used: 0.036 (sec). Leaf size: 62

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

\begin{align*} y(x)&\to (-1)^{-1/n} n^{\frac {1}{n}-1} a^{-1/n} \left (x^n\right )^{-1/n} \left (c_1 (-1)^{\frac {1}{n}} \Gamma \left (\frac {1}{n},0,\frac {a x^n}{n}\right )+c_2 n\right ) \end{align*}

✗ Sympy

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

False