61.31.3 problem 184

Internal problem ID [12605]
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 : 184
Date solved : Thursday, March 13, 2025 at 11:53:00 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.129 (sec). Leaf size: 110

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

\[ y = \frac {-x c_{1} \left (\Gamma \left (a , -\frac {b}{x}\right )-\Gamma \left (a \right )\right ) \left (-1\right )^{-a} \left (a -2\right ) b^{-a +1}-c_{1} \left (\Gamma \left (a , -\frac {b}{x}\right )-\Gamma \left (a \right )\right ) \left (-1\right )^{-a} b^{-a +2}-b \,x^{-a +1} c_{1} {\mathrm e}^{\frac {b}{x}}+c_{2} \left (a -2\right ) x +c_{1} x^{-a +2} {\mathrm e}^{\frac {b}{x}}+c_{2} b}{x} \]

✓ Mathematica. Time used: 0.584 (sec). Leaf size: 82

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

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

✗ Sympy

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