23.3.584 problem 592

Internal problem ID [6298]
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 : 592
Date solved : Tuesday, September 30, 2025 at 02:45:13 PM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 30

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

\[ y = \sin \left (x^{1-a}\right ) c_1 +c_2 \cos \left (x^{1-a} \operatorname {csgn}\left (-1+a \right )\right ) \]

✓ Mathematica. Time used: 0.023 (sec). Leaf size: 29

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

\begin{align*} y(x)&\to c_1 \cos \left (x^{1-a}\right )-c_2 \sin \left (x^{1-a}\right ) \end{align*}

✗ Sympy

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

TypeError : invalid input: 1 - a