59.1.141 problem 143

Internal problem ID [9313]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 143
Date solved : Sunday, March 30, 2025 at 02:32:08 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 4 x^{2} y^{\prime \prime }+2 x^{3} y^{\prime }+\left (3 x^{2}+1\right ) y&=0 \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 25

ode:=4*x^2*diff(diff(y(x),x),x)+2*x^3*diff(y(x),x)+(3*x^2+1)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.193 (sec). Leaf size: 44

ode=4*x^2*D[y[x],{x,2}]+2*x^3*D[y[x],x]+(1+3*x^2)*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✗ Sympy

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

False