2.27.10 Problem 10

Internal problem ID [13671]
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 : 10
Date solved : Friday, December 19, 2025 at 10:20:55 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

2.27.10.1 ✓ Maple. Time used: 0.048 (sec). Leaf size: 89

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

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

Maple trace

Methods for second order ODEs: 
--- Trying classification methods --- 
trying a symmetry of the form [xi=0, eta=F(x)] 
checking if the LODE is missing y 
-> Trying an equivalence, under non-integer power transformations, 
   to LODEs admitting Liouvillian solutions. 
   -> Trying a Liouvillian solution using Kovacics algorithm 
   <- No Liouvillian solutions exists 
-> Trying a solution in terms of special functions: 
   -> Bessel 
   -> elliptic 
   -> Legendre 
   -> Whittaker 
      -> hyper3: Equivalence to 1F1 under a power @ Moebius 
      <- hyper3 successful: received ODE is equivalent to the 1F1 ODE 
   <- Whittaker successful 
<- special function solution successful
 

2.27.10.2 ✓ Mathematica. Time used: 0.165 (sec). Leaf size: 225

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

\begin{align*} y(x)&\to 2^{\frac {n}{2 n+2}} x^{-n/2} \left (x^{n+1}\right )^{\frac {n}{2 n+2}} e^{-\frac {\sqrt {a} x^{n+1}}{\sqrt {-(n+1)^2}}} \left (c_1 \operatorname {HypergeometricU}\left (-\frac {(n+1) \left (n b+b+\sqrt {a} n \sqrt {-(n+1)^2}\right )}{2 \sqrt {a} \left (-(n+1)^2\right )^{3/2}},\frac {n}{n+1},\frac {2 \sqrt {a} x^{n+1}}{\sqrt {-(n+1)^2}}\right )+c_2 L_{\frac {(n+1) \left (n b+b+\sqrt {a} n \sqrt {-(n+1)^2}\right )}{2 \sqrt {a} \left (-(n+1)^2\right )^{3/2}}}^{-\frac {1}{n+1}}\left (\frac {2 \sqrt {a} x^{n+1}}{\sqrt {-(n+1)^2}}\right )\right ) \end{align*}

2.27.10.3 ✗ Sympy

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

TypeError : Mul object cannot be interpreted as an integer