23.3.456 problem 461

Internal problem ID [6170]
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 : 461
Date solved : Tuesday, September 30, 2025 at 02:24:02 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (x^{4}+2 x^{2}+a \right ) y+4 x^{3} y^{\prime }+4 x^{2} y^{\prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 41

ode:=(x^4+2*x^2+a)*y(x)+4*x^3*diff(y(x),x)+4*x^2*diff(diff(y(x),x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.058 (sec). Leaf size: 70

ode=(a + 2*x^2 + x^4)*y[x] + 4*x^3*D[y[x],x] + 4*x^2*D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {e^{-\frac {x^2}{4}} x^{\frac {1}{2}-\frac {\sqrt {1-a}}{2}} \left (c_2 x^{\sqrt {1-a}}+\sqrt {1-a} c_1\right )}{\sqrt {1-a}} \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(4*x**3*Derivative(y(x), x) + 4*x**2*Derivative(y(x), (x, 2)) + (a + x**4 + 2*x**2)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

ValueError : Expected Expr or iterable but got None