55.33.4 problem 214

Internal problem ID [13987]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-7
Problem number : 214
Date solved : Thursday, October 02, 2025 at 09:08:24 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} b y+2 x^{2} \left (x +a \right ) y^{\prime }+x^{4} y^{\prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 42

ode:=x^4*diff(diff(y(x),x),x)+2*x^2*(x+a)*diff(y(x),x)+b*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.023 (sec). Leaf size: 51

ode=x^4*D[y[x],{x,2}]+2*x^2*(x+a)*D[y[x],x]+b*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
y = Function("y") 
ode = Eq(b*y(x) + x**4*Derivative(y(x), (x, 2)) + 2*x**2*(a + x)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (-b*y(x) - x**4*Derivative(y(x), (x, 2)))/(2*x**2*(a + x)) cannot be solved by the factorable group method