23.3.221 problem 223

Internal problem ID [5935]
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 : 223
Date solved : Tuesday, September 30, 2025 at 02:06:27 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} -8 x^{3} y-\left (2 x^{2}+1\right ) y^{\prime }+x y^{\prime \prime }&=4 x^{3} {\mathrm e}^{-x^{2}} \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 28

ode:=-8*x^3*y(x)-(2*x^2+1)*diff(y(x),x)+x*diff(diff(y(x),x),x) = 4*x^3/exp(x^2); 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.042 (sec). Leaf size: 38

ode=-8*x^3*y[x] - (1 + 2*x^2)*D[y[x],x] + x*D[y[x],{x,2}] == (4*x^3)/E^(x^2); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{9} e^{-x^2} \left (-3 x^2+9 c_1 e^{3 x^2}-1+9 c_2\right ) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-8*x**3*y(x) - 4*x**3*exp(-x**2) + x*Derivative(y(x), (x, 2)) - (2*x**2 + 1)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -x*(-8*x**2*y(x)*exp(x**2) - 4*x**2 + exp(x**2)*Derivative(y(x),