55.29.29 problem 89

Internal problem ID [13862]
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 : 89
Date solved : Thursday, October 02, 2025 at 08:07:58 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.045 (sec). Leaf size: 48

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

\[ y = {\mathrm e}^{-a x} \left (a x +1\right ) \left (c_2 \int \frac {x \,{\mathrm e}^{-\frac {\left (a b \,x^{2}+\frac {3}{2} b x -3 a \right ) x}{3}}}{\left (a x +1\right )^{2}}d x +c_1 \right ) \]

✓ Mathematica. Time used: 1.927 (sec). Leaf size: 72

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

\begin{align*} y(x)&\to \frac {e^{-a x} (a x+1) \left (c_2 \int _1^x\frac {a^2 \exp \left (-\frac {1}{6} K[1] \left (3 b K[1]+2 a \left (b K[1]^2-3\right )\right )\right ) K[1]}{(a K[1]+1)^2}dK[1]+c_1\right )}{a} \end{align*}

✗ Sympy

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

False