55.30.29 problem 138

Internal problem ID [13911]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-4
Problem number : 138
Date solved : Friday, October 03, 2025 at 06:55:37 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 48

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

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

✓ Mathematica. Time used: 0.315 (sec). Leaf size: 64

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

\begin{align*} y(x)&\to e^{-k x} x^{-n} \left (c_1-c_2 x^{-b+2 n+1} (x (a-2 k))^{b-2 n-1} \Gamma (-b+2 n+1,(a-2 k) x)\right ) \end{align*}

✗ Sympy

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

ValueError : Expected Expr or iterable but got None