61.33.2 problem 239

Internal problem ID [12661]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-8. Other equations.
Problem number : 239
Date solved : Monday, March 31, 2025 at 06:49:49 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{6} y^{\prime \prime }+\left (3 x^{2}+a \right ) x^{3} y^{\prime }+b y&=0 \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 43

ode:=x^6*diff(diff(y(x),x),x)+(3*x^2+a)*x^3*diff(y(x),x)+b*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.05 (sec). Leaf size: 56

ode=x^6*D[y[x],{x,2}]+(3*x^2+a)*x^3*D[y[x],x]+b*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to e^{\frac {a-\sqrt {a^2-4 b}}{4 x^2}} \left (c_1 e^{\frac {\sqrt {a^2-4 b}}{2 x^2}}+c_2\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
y = Function("y") 
ode = Eq(b*y(x) + x**6*Derivative(y(x), (x, 2)) + x**3*(a + 3*x**2)*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**6*Derivative(y(x), (x, 2)))/(x**3*(a + 3*x**2)) cannot be solved by the factorable group method