55.34.11 problem 249

Internal problem ID [14022]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-8. Other equations.
Problem number : 249
Date solved : Thursday, October 02, 2025 at 09:09:06 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✗ Maple

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

\[ \text {No solution found} \]

✗ Mathematica

ode=(a*x^n+b)*D[y[x],{x,2}]+(c*x^n+d)*D[y[x],x]+\[Lambda]*((c-a*\[Lambda])*x^n+d-b*\[Lambda])*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") 
c = symbols("c") 
d = symbols("d") 
lambda_ = symbols("lambda_") 
n = symbols("n") 
y = Function("y") 
ode = Eq(lambda_*(-b*lambda_ + d + x**n*(-a*lambda_ + c))*y(x) + (a*x**n + b)*Derivative(y(x), (x, 2)) + (c*x**n + d)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

TypeError : Symbol object cannot be interpreted as an integer