59.1.737 problem 757

Internal problem ID [9909]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 757
Date solved : Wednesday, March 05, 2025 at 08:00:46 AM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

\begin{align*} x \left (x +2\right ) y^{\prime \prime }+\left (x +1\right ) y^{\prime }-4 y&=0 \end{align*}

✓ Maple. Time used: 0.007 (sec). Leaf size: 28

ode:=x*(x+2)*diff(diff(y(x),x),x)+(1+x)*diff(y(x),x)-4*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_{2} \left (x +1\right ) \sqrt {x \left (x +2\right )}+2 \left (x^{2}+2 x +\frac {1}{2}\right ) c_{1} \]

✓ Mathematica. Time used: 0.061 (sec). Leaf size: 45

ode=x*(x+2)*D[y[x],{x,2}]+(x+1)*D[y[x],x]-4*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to c_1 \cosh \left (4 \text {arctanh}\left (\frac {1}{\sqrt {\frac {x}{x+2}}}\right )\right )+i c_2 \sinh \left (4 \text {arctanh}\left (\frac {1}{\sqrt {\frac {x}{x+2}}}\right )\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*(x + 2)*Derivative(y(x), (x, 2)) + (x + 1)*Derivative(y(x), x) - 4*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False