61.33.23 problem 261

Internal problem ID [12682]
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 : 261
Date solved : Wednesday, March 05, 2025 at 08:16:26 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 13.873 (sec). Leaf size: 68

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

\[ y = -\left (\left (\int {\mathrm e}^{\int \frac {-2 a \,x^{n}-2 x^{m} b -2 c +x^{2}-2 \lambda x +\lambda ^{2}}{\left (a \,x^{n}+x^{m} b +c \right ) \left (-\lambda +x \right )}d x}d x \right ) c_{1} +c_{2} \right ) \left (\lambda -x \right ) \]

✗ Mathematica

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

TypeError : Symbol object cannot be interpreted as an integer