62.31.5 problem Ex 5

Internal problem ID [12906]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter VIII, Linear differential equations of the second order. Article 54. Change of independent variable. Page 127
Problem number : Ex 5
Date solved : Monday, March 31, 2025 at 07:24:12 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 28

ode:=x*diff(diff(y(x),x),x)-(2*x^2+1)*diff(y(x),x)-8*x^3*y(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: 0.075 (sec). Leaf size: 38

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

\[ y(x)\to \frac {1}{9} e^{-x^2} \left (-3 x^2+9 c_1 e^{3 x^2}-1+9 c_2\right ) \]

✗ 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), (x, 2)))*exp(-x**2)/(2*x**2 + 1) + Derivative(y(x), x) cannot be solved by the factorable group method