55.28.5 problem 15

Internal problem ID [13788]
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 : 15
Date solved : Thursday, October 02, 2025 at 08:06:44 AM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 30

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

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

✓ Mathematica. Time used: 0.563 (sec). Leaf size: 46

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

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

False