55.33.22 problem 231

Internal problem ID [14005]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-7
Problem number : 231
Date solved : Thursday, October 02, 2025 at 09:08:44 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✗ Maple

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

\[ \text {No solution found} \]

✗ Mathematica

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

Not solved

✗ Sympy

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

TypeError : Add object cannot be interpreted as an integer