23.3.585 problem 593

Internal problem ID [6299]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 3. THE DIFFERENTIAL EQUATION IS LINEAR AND OF SECOND ORDER, page 311
Problem number : 593
Date solved : Tuesday, September 30, 2025 at 02:45:16 PM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} a^{2} x^{-1+a} y+\left (1-2 a \right ) x^{a} y^{\prime }+x^{1+a} y^{\prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 14

ode:=a^2*x^(a-1)*y(x)+(1-2*a)*x^a*diff(y(x),x)+x^(a+1)*diff(diff(y(x),x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.015 (sec). Leaf size: 18

ode=a^2*x^(-1 + a)*y[x] + (1 - 2*a)*x^a*D[y[x],x] + x^(1 + a)*D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to x^a (a c_2 \log (x)+c_1) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(a**2*x**(a - 1)*y(x) + x**a*(1 - 2*a)*Derivative(y(x), x) + x**(a + 1)*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

TypeError : invalid input: 2*a