59.1.60 problem 62

Internal problem ID [9232]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 62
Date solved : Sunday, March 30, 2025 at 02:26:09 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.011 (sec). Leaf size: 73

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

\[ y = c_2 \,{\mathrm e}^{-\frac {\left (x -3\right )^{2}}{2}} \left (\operatorname {erf}\left (\frac {\sqrt {2}\, \sqrt {-\left (x -3\right )^{2}}}{2}\right )-1\right ) \left (x -4\right ) \left (x -2\right ) \sqrt {\pi }-\sqrt {2}\, \sqrt {-\left (x -3\right )^{2}}\, c_2 -c_1 \,{\mathrm e}^{-\frac {\left (x -3\right )^{2}}{2}} \left (x -2\right ) \left (x -4\right ) \]

✓ Mathematica. Time used: 0.69 (sec). Leaf size: 62

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

\[ y(x)\to e^{-\frac {1}{2} (x-6) x} \left (x^2-6 x+8\right ) \left (c_2 \int _1^x\frac {e^{\frac {1}{2} (K[1]-6) K[1]}}{(K[1]-4)^2 (K[1]-2)^2}dK[1]+c_1\right ) \]

✗ Sympy

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

False