59.1.45 problem 47

Internal problem ID [9217]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 47
Date solved : Sunday, March 30, 2025 at 02:25:47 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 71

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

\[ y = c_1 \,{\mathrm e}^{-x} \left (x^{2}-6 x +4\right ) \left (2+x \right )^{3}+c_2 \left ({\mathrm e}^{-2-x} \left (x^{2}-6 x +4\right ) \left (2+x \right )^{3} \operatorname {Ei}_{1}\left (-2-x \right )+x^{4}-x^{3}-18 x^{2}-22 x +8\right ) \]

✓ Mathematica. Time used: 0.688 (sec). Leaf size: 106

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

\[ y(x)\to \left (x^2-6 x+4\right ) \exp \left (\int _1^x\left (\frac {2}{K[1]+2}-\frac {1}{2}\right )dK[1]-\frac {1}{2} \int _1^x\frac {K[2]}{K[2]+2}dK[2]\right ) \left (c_2 \int _1^x\frac {\exp \left (-2 \int _1^{K[3]}\left (\frac {2}{K[1]+2}-\frac {1}{2}\right )dK[1]\right )}{\left (K[3]^2-6 K[3]+4\right )^2}dK[3]+c_1\right ) \]

✗ Sympy

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

False