61.28.8 problem 68

Internal problem ID [12489]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-3
Problem number : 68
Date solved : Thursday, March 13, 2025 at 11:42:52 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 2.178 (sec). Leaf size: 166

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

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

✗ Mathematica

ode=x*D[y[x],{x,2}]+a*D[y[x],x]+b*x^n*(-b*x^(n+1)+a+n)*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*Derivative(y(x), x) + b*x**n*(a - b*x**(n + 1) + n)*y(x) + x*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

ValueError : Expected Expr or iterable but got None