61.33.18 problem 256

Internal problem ID [12677]
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 : 256
Date solved : Monday, March 31, 2025 at 06:50:44 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.198 (sec). Leaf size: 53

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

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

✓ Mathematica. Time used: 60.196 (sec). Leaf size: 106

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

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

✗ Sympy

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

TypeError : Add object cannot be interpreted as an integer