23.3.223 problem 225

Internal problem ID [5937]
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 : 225
Date solved : Tuesday, September 30, 2025 at 02:06:29 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} a \,x^{2} \left (a \,x^{3}+1\right ) y-\left (-2 a \,x^{3}+1\right ) y^{\prime }+x y^{\prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 19

ode:=a*x^2*(a*x^3+1)*y(x)-(-2*a*x^3+1)*diff(y(x),x)+x*diff(diff(y(x),x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = {\mathrm e}^{-\frac {a \,x^{3}}{3}} \left (c_2 \,x^{2}+c_1 \right ) \]

✓ Mathematica. Time used: 0.021 (sec). Leaf size: 30

ode=a*x^2*(1 + a*x^3)*y[x] - (1 - 2*a*x^3)*D[y[x],x] + x*D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{2} e^{-\frac {a x^3}{3}} \left (c_2 x^2+2 c_1\right ) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(a*x**2*(a*x**3 + 1)*y(x) + x*Derivative(y(x), (x, 2)) - (-2*a*x**3 + 1)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False