55.28.33 problem 43

Internal problem ID [13816]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-2
Problem number : 43
Date solved : Thursday, October 02, 2025 at 08:07:09 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.058 (sec). Leaf size: 41

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

\[ y = {\mathrm e}^{-a x} \left (a x +1\right ) \left (c_2 \int \frac {{\mathrm e}^{-\frac {x^{3} b \left (a x +\frac {4}{3}\right )}{4}}}{\left (a x +1\right )^{2}}d x +c_1 \right ) \]

✓ Mathematica. Time used: 2.465 (sec). Leaf size: 57

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

\begin{align*} y(x)&\to e^{-a x} (a x+1) \left (c_2 \int _1^x\frac {e^{-\frac {1}{12} b K[1]^3 (3 a K[1]+4)}}{(a K[1]+1)^2}dK[1]+c_1\right ) \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 + 1)*y(x) + (a*b*x**3 + 2*a + b*x**2)*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False