61.31.11 problem 192

Internal problem ID [12613]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-6
Problem number : 192
Date solved : Monday, March 31, 2025 at 05:54:03 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 20

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

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

✓ Mathematica. Time used: 0.199 (sec). Leaf size: 44

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

\[ y(x)\to (c_2 x+c_1) \exp \left (-\frac {1}{2} \int _1^x\left (\frac {2 a}{b+a K[1]}-\frac {4}{K[1]}\right )dK[1]\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
y = Function("y") 
ode = Eq(x**2*(a*x + b)*Derivative(y(x), (x, 2)) - 2*x*(a*x + 2*b)*Derivative(y(x), x) + (2*a*x + 6*b)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False