23.3.434 problem 439

Internal problem ID [6148]
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 : 439
Date solved : Tuesday, September 30, 2025 at 02:22:22 PM
CAS classification : [_Jacobi, [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 83

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

\[ y = c_1 \sin \left (\frac {\sqrt {\left (-1+x \right ) x}\, \left (-\ln \left (2\right )+\ln \left (-1+2 x +2 \sqrt {\left (-1+x \right ) x}\right )\right )}{\sqrt {-1+x}\, \sqrt {x}}\right )+c_2 \cos \left (\frac {\sqrt {\left (-1+x \right ) x}\, \left (-\ln \left (2\right )+\ln \left (-1+2 x +2 \sqrt {\left (-1+x \right ) x}\right )\right )}{\sqrt {-1+x}\, \sqrt {x}}\right ) \]

✓ Mathematica. Time used: 0.078 (sec). Leaf size: 42

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

\begin{align*} y(x)&\to c_1 \cos \left (2 \text {arctanh}\left (\frac {1}{\sqrt {\frac {x-1}{x}}}\right )\right )+c_2 \sin \left (2 \text {arctanh}\left (\frac {1}{\sqrt {\frac {x-1}{x}}}\right )\right ) \end{align*}

✗ Sympy

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

False