61.33.26 problem 264

Internal problem ID [12685]
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 : 264
Date solved : Monday, March 31, 2025 at 06:51:15 AM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

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

✓ Maple. Time used: 0.038 (sec). Leaf size: 41

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

\[ y = {\mathrm e}^{-\int \left (a \,x^{n}+b \right )^{-m}d x} \left (c_2 \int {\mathrm e}^{\int \left (a \,x^{n}+b \right )^{-m}d x}d x +c_1 \right ) \]

✓ Mathematica. Time used: 60.096 (sec). Leaf size: 116

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

\[ y(x)\to \exp \left (-x \left (a x^n+b\right )^{-m} \left (\frac {a x^n}{b}+1\right )^m \operatorname {Hypergeometric2F1}\left (m,\frac {1}{n},1+\frac {1}{n},-\frac {a x^n}{b}\right )\right ) \left (\int _1^x\exp \left (\operatorname {Hypergeometric2F1}\left (m,\frac {1}{n},1+\frac {1}{n},-\frac {a K[1]^n}{b}\right ) K[1] \left (a K[1]^n+b\right )^{-m} \left (\frac {a K[1]^n}{b}+1\right )^m\right ) c_1dK[1]+c_2\right ) \]

✗ Sympy

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

TypeError : Add object cannot be interpreted as an integer