61.27.41 problem 51

Internal problem ID [12472]
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 : 51
Date solved : Monday, March 31, 2025 at 05:35:43 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.127 (sec). Leaf size: 135

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

\[ y = c_1 x \,{\mathrm e}^{-x b}+\frac {c_2 \,x^{-\frac {3 n}{2}} \left (\left (n +1\right ) \left (a \,x^{n} x +n \right ) \operatorname {WhittakerM}\left (\frac {-2-n}{2+2 n}, \frac {2 n +1}{2+2 n}, \frac {a \,x^{n} x}{n +1}\right )+n^{2} \operatorname {WhittakerM}\left (\frac {n}{2+2 n}, \frac {2 n +1}{2+2 n}, \frac {a \,x^{n} x}{n +1}\right )\right ) {\mathrm e}^{-\frac {\left (a \,x^{n}+2 b \left (n +1\right )\right ) x}{2+2 n}}}{x} \]

✗ Mathematica

ode=D[y[x],{x,2}]+(a*x^n+2*b)*D[y[x],x]+(a*b*x^n-a*x^(n-1)+b^2)*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") 
n = symbols("n") 
y = Function("y") 
ode = Eq((a*x**n + 2*b)*Derivative(y(x), x) + (a*b*x**n - a*x**(n - 1) + b**2)*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