53.1.496 problem 512

Internal problem ID [10968]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 512
Date solved : Tuesday, September 30, 2025 at 07:34:36 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.010 (sec). Leaf size: 49

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

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

✓ Mathematica. Time used: 0.12 (sec). Leaf size: 50

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

\begin{align*} y(x)&\to e^{-x^2} \left (x^2-1\right ) \left (c_2 \int _1^x\frac {e^{\frac {K[1]^2}{2}}}{\left (K[1]^2-1\right )^2}dK[1]+c_1\right ) \end{align*}

✗ Sympy

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

False