61.29.14 problem 123

Internal problem ID [12544]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-4
Problem number : 123
Date solved : Monday, March 31, 2025 at 05:38:30 AM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} x^{2} y^{\prime \prime }+a x y^{\prime }+b y&=0 \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 52

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

\[ y = x^{-\frac {a}{2}} \sqrt {x}\, \left (x^{\frac {\sqrt {a^{2}-2 a -4 b +1}}{2}} c_1 +x^{-\frac {\sqrt {a^{2}-2 a -4 b +1}}{2}} c_2 \right ) \]

✓ Mathematica. Time used: 0.044 (sec). Leaf size: 57

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

\[ y(x)\to x^{\frac {1}{2} \left (-\sqrt {a^2-2 a-4 b+1}-a+1\right )} \left (c_2 x^{\sqrt {a^2-2 a-4 b+1}}+c_1\right ) \]

✓ Sympy. Time used: 2.208 (sec). Leaf size: 617

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

\[ \text {Solution too large to show} \]