59.1.185 problem 187

Internal problem ID [9357]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 187
Date solved : Sunday, March 30, 2025 at 02:33:06 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 39

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

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

✓ Mathematica. Time used: 0.114 (sec). Leaf size: 46

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

\[ y(x)\to \frac {c_2 \left (e^{x^2} x^4 \operatorname {ExpIntegralEi}\left (-x^2\right )+x^2-1\right )}{4 x^2}+c_1 e^{x^2} x^2 \]

✗ Sympy

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

False