23.3.125 problem 127

Internal problem ID [5839]
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 : 127
Date solved : Tuesday, September 30, 2025 at 02:03:58 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} -2 a \left (-2 a \,x^{2}+1\right ) y-4 a x y^{\prime }+y^{\prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 16

ode:=-2*a*(-2*a*x^2+1)*y(x)-4*a*x*diff(y(x),x)+diff(diff(y(x),x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = {\mathrm e}^{a \,x^{2}} \left (c_2 x +c_1 \right ) \]

✓ Mathematica. Time used: 0.015 (sec). Leaf size: 20

ode=-2*a*(1 - 2*a*x^2)*y[x] - 4*a*x*D[y[x],x] + D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to e^{a x^2} (c_2 x+c_1) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(-4*a*x*Derivative(y(x), x) - 2*a*(-2*a*x**2 + 1)*y(x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False