2.28.37 Problem 47

Internal problem ID [13708]
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 : 47
Date solved : Friday, December 19, 2025 at 10:57:40 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

2.28.37.1 ✓ Maple. Time used: 0.052 (sec). Leaf size: 171

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

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

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
 

2.28.37.2 ✓ Mathematica. Time used: 0.175 (sec). Leaf size: 333

ode=D[y[x],{x,2}]+a*x^n*D[y[x],x]+(b*x^(2*n)+c*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}} \exp \left (-\frac {1}{2} x^{n+1} \left (\frac {\sqrt {a^2-4 b}}{\sqrt {(n+1)^2}}+\frac {a}{n+1}\right )\right ) \left (c_1 \operatorname {HypergeometricU}\left (\frac {n \left (\sqrt {(n+1)^2} a^2+\sqrt {a^2-4 b} (n+1) a-4 b \sqrt {(n+1)^2}\right )-2 \sqrt {a^2-4 b} c (n+1)}{2 \left (a^2-4 b\right ) (n+1) \sqrt {(n+1)^2}},\frac {n}{n+1},\frac {\sqrt {a^2-4 b} x^{n+1}}{\sqrt {(n+1)^2}}\right )+c_2 L_{\frac {2 \sqrt {a^2-4 b} c (n+1)-n \left (\sqrt {(n+1)^2} a^2+\sqrt {a^2-4 b} (n+1) a-4 b \sqrt {(n+1)^2}\right )}{2 \left (a^2-4 b\right ) (n+1) \sqrt {(n+1)^2}}}^{-\frac {1}{n+1}}\left (\frac {\sqrt {a^2-4 b} x^{n+1}}{\sqrt {(n+1)^2}}\right )\right ) \end{align*}

2.28.37.3 ✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
c = symbols("c") 
n = symbols("n") 
y = Function("y") 
ode = Eq(a*x**n*Derivative(y(x), x) + (b*x**(2*n) + c*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