2.32.17 Problem 198

Internal problem ID [13859]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-6
Problem number : 198
Date solved : Friday, December 19, 2025 at 04:32:36 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (a \,x^{3}+b \,x^{2}+c x \right ) y^{\prime \prime }+\left (x^{2} n +m x +k \right ) y^{\prime }+\left (k -1\right ) \left (\left (-a k +n \right ) x +m -b k \right ) y&=0 \\ \end{align*}

2.32.17.1 ✗ Maple

ode:=(a*x^3+b*x^2+c*x)*diff(diff(y(x),x),x)+(n*x^2+m*x+k)*diff(y(x),x)+(-1+k)*((-a*k+n)*x+m-b*k)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ \text {No solution found} \]

2.32.17.2 ✓ Mathematica. Time used: 117.68 (sec). Leaf size: 598

ode=(a*x^3+b*x^2+c*x)*D[y[x],{x,2}]+(n*x^2+m*x+k)*D[y[x],x]+(k-1)*((n-a*k)*x+m-b*k)*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\frac {2^{-\frac {k}{c}} \left (-\frac {a x}{\sqrt {b^2-4 a c}+b}\right )^{1-\frac {k}{c}} \left (a c_1 x \left (-2^{\frac {k}{c}}\right ) \left (-\frac {a x}{\sqrt {b^2-4 a c}+b}\right )^{\frac {k}{c}-1} \text {HeunG}\left [\frac {b-\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}+b},-\frac {2 (k-1) (b k-m)}{\sqrt {b^2-4 a c}+b},\frac {a \left (\sqrt {\frac {(-2 a k+a+n)^2}{a^2}}-1\right )+n}{2 a},\frac {n-a \left (\sqrt {\frac {(-2 a k+a+n)^2}{a^2}}+1\right )}{2 a},\frac {k}{c},\frac {2 a^2 k-a \left (m \sqrt {b^2-4 a c}+b m+2 c n\right )+b n \left (\sqrt {b^2-4 a c}+b\right )}{a \left (b \sqrt {b^2-4 a c}-4 a c+b^2\right )},-\frac {2 a x}{\sqrt {b^2-4 a c}+b}\right ]-2 a c_2 x \text {HeunG}\left [\frac {b-\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}+b},-\frac {2 (c-1) k \sqrt {b^2-4 a c} (b (c (k-1)+k)-c m)}{c^2 \left (b \sqrt {b^2-4 a c}-4 a c+b^2\right )},\frac {a \left (-\sqrt {\frac {(-2 a k+a+n)^2}{a^2}}\right )+a+n}{2 a}-\frac {k}{c},\frac {a \left (c \sqrt {\frac {(-2 a k+a+n)^2}{a^2}}+c-2 k\right )+c n}{2 a c},2-\frac {k}{c},\frac {2 a^2 k-a \left (m \sqrt {b^2-4 a c}+b m+2 c n\right )+b n \left (\sqrt {b^2-4 a c}+b\right )}{a \left (b \sqrt {b^2-4 a c}-4 a c+b^2\right )},-\frac {2 a x}{\sqrt {b^2-4 a c}+b}\right ]\right )}{a x} \end{align*}

2.32.17.3 ✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
c = symbols("c") 
k = symbols("k") 
m = symbols("m") 
n = symbols("n") 
y = Function("y") 
ode = Eq((k - 1)*(-b*k + m + x*(-a*k + n))*y(x) + (k + m*x + n*x**2)*Derivative(y(x), x) + (a*x**3 + b*x**2 + c*x)*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

ValueError : Expected Expr or iterable but got None
 

Python version: 3.12.3 (main, Aug 14 2025, 17:47:21) [GCC 13.3.0] 
Sympy version 1.14.0
 

classify_ode(ode,func=y(x)) 
 
('factorable', '2nd_power_series_regular')