59.1.70 problem 72

Internal problem ID [9242]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 72
Date solved : Sunday, March 30, 2025 at 02:30:28 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 58

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

\[ y = c_1 \,{\mathrm e}^{-x} x \left (x -4\right ) \left (2+x \right )^{2}+c_2 \left ({\mathrm e}^{-2-x} x \left (x -4\right ) \left (2+x \right )^{2} \operatorname {Ei}_{1}\left (-2-x \right )+x^{3}-x^{2}-10 x -6\right ) \]

✓ Mathematica. Time used: 0.542 (sec). Leaf size: 106

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

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

✗ Sympy

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

False