55.32.18 problem 199

Internal problem ID [13972]
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 : 199
Date solved : Thursday, October 02, 2025 at 08:22:12 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✗ Maple

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

\[ \text {No solution found} \]

✓ Mathematica. Time used: 12.165 (sec). Leaf size: 192

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

\begin{align*} y(x)&\to \frac {\left (x (a x+2 b+m x-1)+c (2 b+4 m x-1)+4 c^2 m\right ) \left (c_2 \int _1^x\frac {\exp \left (\frac {(b m-2 a (b+2 c m-1)) \arctan \left (\frac {b+2 a K[1]}{\sqrt {4 a c-b^2}}\right )}{a \sqrt {4 a c-b^2}}\right ) K[1] (c+K[1] (b+a K[1]))^{-\frac {m}{2 a}}}{\left (4 m c^2+(2 b+4 m K[1]-1) c+K[1] (2 b+a K[1]+m K[1]-1)\right )^2}dK[1]+c_1\right )}{a+2 b (c+1)+4 c^2 m+4 c m-c+m-1} \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
c = symbols("c") 
m = symbols("m") 
y = Function("y") 
ode = Eq((-2*m*x + 1)*y(x) + (-c + x**2*(-a + m) + x*(2*c*m - 1))*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)
 

False