61.31.18 problem 199

Internal problem ID [12620]
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 : Monday, March 31, 2025 at 05:54:33 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: 11.161 (sec). Leaf size: 277

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]
 

\[ y(x)\to \frac {\left (x (a x+2 b+m x-1)+c (2 b+4 m x-1)+4 c^2 m\right ) \left (c_2 \int _1^x\exp \left (\int _1^{K[2]}\frac {4 m (1-2 m K[1]) c^3+\left (-12 m^2 K[1]^2+2 m (2 a K[1]+1) K[1]+b (2-4 m K[1])-1\right ) c^2-2 K[1]^2 (a (-b+3 m K[1]+2)+m (7 b+3 m K[1]-2)) c-K[1]^2 \left (4 b^2+(6 a K[1]+6 m K[1]-4) b+3 a^2 K[1]^2+(m K[1]-1)^2+2 a K[1] (2 m K[1]-1)\right )}{K[1] (c+K[1] (b+a K[1])) \left (4 m c^2+(2 b+4 m K[1]-1) c+K[1] (2 b+a K[1]+m K[1]-1)\right )}dK[1]\right )dK[2]+c_1\right )}{a+2 b (c+1)+4 c^2 m+4 c m-c+m-1} \]

✗ 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