61.27.35 problem 45

Internal problem ID [12466]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-2
Problem number : 45
Date solved : Thursday, March 13, 2025 at 11:42:42 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 1.585 (sec). Leaf size: 96

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

\[ y = x \left (\operatorname {KummerU}\left (\frac {1+n -\frac {b}{a}}{n +1}, \frac {n +2}{n +1}, \frac {a \,x^{n} x}{n +1}\right ) c_{2} +\operatorname {KummerM}\left (\frac {1+n -\frac {b}{a}}{n +1}, \frac {n +2}{n +1}, \frac {a \,x^{n} x}{n +1}\right ) c_{1} \right ) {\mathrm e}^{-\frac {a \,x^{n} x}{n +1}} \]

✓ Mathematica. Time used: 0.091 (sec). Leaf size: 120

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

\[ y(x)\to c_2 \left (\frac {1}{n}+1\right )^{-\frac {1}{n+1}} n^{-\frac {1}{n+1}} a^{\frac {1}{n+1}} \left (x^n\right )^{\frac {1}{n}} \operatorname {Hypergeometric1F1}\left (\frac {a+b}{n a+a},\frac {n+2}{n+1},-\frac {a \left (x^n\right )^{1+\frac {1}{n}}}{n+1}\right )+c_1 \operatorname {Hypergeometric1F1}\left (\frac {b}{n a+a},\frac {n}{n+1},-\frac {a \left (x^n\right )^{1+\frac {1}{n}}}{n+1}\right ) \]

✗ Sympy

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

TypeError : Add object cannot be interpreted as an integer