59.1.78 problem 80

Internal problem ID [9250]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 80
Date solved : Sunday, March 30, 2025 at 02:30:39 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-3 x y^{\prime }+\left (2 x^{2}+5\right ) y&=0 \end{align*}

✓ Maple. Time used: 0.012 (sec). Leaf size: 62

ode:=diff(diff(y(x),x),x)-3*x*diff(y(x),x)+(2*x^2+5)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \left (x^{6}-15 x^{4}+45 x^{2}-15\right ) \left (\operatorname {erfi}\left (\frac {\sqrt {2}\, x}{2}\right ) \sqrt {2}\, \sqrt {\pi }\, c_1 +c_2 \right ) {\mathrm e}^{\frac {x^{2}}{2}}-2 \,{\mathrm e}^{x^{2}} c_1 x \left (x^{2}-11\right ) \left (x^{2}-3\right ) \]

✓ Mathematica. Time used: 0.871 (sec). Leaf size: 74

ode=D[y[x],{x,2}]-3*x*D[y[x],x]+(5+2*x^2)*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to e^{\frac {x^2}{2}} \left (x^6-15 x^4+45 x^2-15\right ) \left (c_2 \int _1^x\frac {e^{\frac {K[1]^2}{2}}}{\left (K[1]^6-15 K[1]^4+45 K[1]^2-15\right )^2}dK[1]+c_1\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-3*x*Derivative(y(x), x) + (2*x**2 + 5)*y(x) + Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False