2.28.44 Problem 54

Internal problem ID [13715]
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 : 54
Date solved : Friday, December 19, 2025 at 11:05:28 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

2.28.44.1 ✓ Maple. Time used: 0.057 (sec). Leaf size: 78

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

\[ y = -\left (c +x \right ) \left (\int \frac {{\mathrm e}^{-\frac {\left (x^{2} a \left (n +2\right ) \left (n +1\right )+x \left (a c +b \right ) \left (n +3\right ) \left (n +1\right )+b c \left (n +3\right ) \left (n +2\right )\right ) x^{n +1}}{\left (n +1\right ) \left (n +2\right ) \left (n +3\right )}}}{\left (c +x \right )^{2}}d x c_1 +c_2 \right ) \]

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 solution in terms of special functions: 
   -> Bessel 
   -> elliptic 
   -> Legendre 
   -> Kummer 
      -> hyper3: Equivalence to 1F1 under a power @ Moebius 
   -> hypergeometric 
      -> heuristic approach 
      -> hyper3: Equivalence to 2F1, 1F1 or 0F1 under a power @ Moebius 
   -> Mathieu 
      -> Equivalence to the rational form of Mathieu ODE under a power @ Moebi\ 
us 
-> Heun: Equivalence to the GHE or one of its 4 confluent cases under a power \ 
@ Moebius 
-> Heun: Equivalence to the GHE or one of its 4 confluent cases under a power \ 
@ Moebius 
-> trying a solution of the form r0(x) * Y + r1(x) * Y where Y = exp(int(r(x),\ 
 dx)) * 2F1([a1, a2], [b1], f) 
-> Trying changes of variables to rationalize or make the ODE simpler 
<- unable to find a useful change of variables 
   trying a symmetry of the form [xi=0, eta=F(x)] 
   trying 2nd order exact linear 
   trying symmetries linear in x and y(x) 
   <- linear symmetries successful
 

2.28.44.2 ✗ Mathematica

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

Not solved

2.28.44.3 ✗ Sympy

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

TypeError : Add object cannot be interpreted as an integer