55.29.16 problem 76

Internal problem ID [13849]
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 : 76
Date solved : Thursday, October 02, 2025 at 08:07:45 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 39

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

\[ y = c_1 \,{\mathrm e}^{a x +\frac {x^{2} \sqrt {-b}}{2}}+c_2 \,{\mathrm e}^{a x -\frac {x^{2} \sqrt {-b}}{2}} \]

✓ Mathematica. Time used: 0.107 (sec). Leaf size: 59

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

\begin{align*} y(x)&\to \frac {1}{2} e^{a x-\frac {1}{2} i \sqrt {b} x^2} \left (2 c_1-\frac {i c_2 e^{i \sqrt {b} x^2}}{\sqrt {b}}\right ) \end{align*}

✗ Sympy

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

False