2.29.39 Problem 99

Internal problem ID [13760]
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 : 99
Date solved : Friday, December 19, 2025 at 11:59:51 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

2.29.39.1 ✓ Maple. Time used: 0.021 (sec). Leaf size: 156

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

\[ y = {\mathrm e}^{-\frac {x^{n} \left (a +\sqrt {a^{2}-4 c}\right )}{2 n}} \left (\operatorname {KummerU}\left (\frac {\left (b +n -1\right ) \sqrt {a^{2}-4 c}+\left (b +n -1\right ) a -2 d}{2 \sqrt {a^{2}-4 c}\, n}, \frac {b +n -1}{n}, \frac {\sqrt {a^{2}-4 c}\, x^{n}}{n}\right ) c_2 +\operatorname {KummerM}\left (\frac {\left (b +n -1\right ) \sqrt {a^{2}-4 c}+\left (b +n -1\right ) a -2 d}{2 \sqrt {a^{2}-4 c}\, n}, \frac {b +n -1}{n}, \frac {\sqrt {a^{2}-4 c}\, x^{n}}{n}\right ) c_1 \right ) \]

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 
   -> Kummer 
      -> hyper3: Equivalence to 1F1 under a power @ Moebius 
      <- hyper3 successful: received ODE is equivalent to the 1F1 ODE 
   <- Kummer successful 
<- special function solution successful
 

2.29.39.2 ✓ Mathematica. Time used: 0.683 (sec). Leaf size: 255

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

\begin{align*} y(x)&\to 2^{\frac {b+n-1}{2 n}} x^{\frac {1}{2}-\frac {n}{2}} \left (x^n\right )^{\frac {n-1}{2 n}} e^{-\frac {\left (\sqrt {a^2-4 c}+a\right ) x^n}{2 n}} \left (c_1 \operatorname {HypergeometricU}\left (\frac {(b+n-1) a^2+\sqrt {a^2-4 c} (b+n-1) a-2 \sqrt {a^2-4 c} d-4 c (b+n-1)}{2 \left (a^2-4 c\right ) n},\frac {b+n-1}{n},\frac {\sqrt {a^2-4 c} x^n}{n}\right )+c_2 L_{-\frac {(b+n-1) a^2+\sqrt {a^2-4 c} (b+n-1) a-2 \sqrt {a^2-4 c} d-4 c (b+n-1)}{2 \left (a^2-4 c\right ) n}}^{\frac {b-1}{n}}\left (\frac {\sqrt {a^2-4 c} x^n}{n}\right )\right ) \end{align*}

2.29.39.3 ✗ Sympy

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

ValueError : Expected Expr or iterable but got None