2.28.39 Problem 49

Internal problem ID [13710]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-2
Problem number : 49
Date solved : Friday, December 19, 2025 at 10:59:30 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

2.28.39.1 ✓ Maple. Time used: 0.050 (sec). Leaf size: 147

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

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

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
 

2.28.39.2 ✓ Mathematica. Time used: 0.116 (sec). Leaf size: 236

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

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

2.28.39.3 ✗ Sympy

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

RecursionError : maximum recursion depth exceeded