55.33.3 problem 213

Internal problem ID [13986]
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 : 213
Date solved : Thursday, October 02, 2025 at 09:08:24 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 25

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

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

✓ Mathematica. Time used: 0.12 (sec). Leaf size: 37

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

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

✗ Sympy

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

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