61.26.9 problem 9

Internal problem ID [12430]
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 Equations Containing Power Functions. page 213
Problem number : 9
Date solved : Thursday, March 13, 2025 at 11:42:21 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.707 (sec). Leaf size: 113

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

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

✗ Mathematica

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

Not solved

✗ Sympy

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

TypeError : Add object cannot be interpreted as an integer