59.1.554 problem 570

Internal problem ID [9726]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 570
Date solved : Sunday, March 30, 2025 at 02:45:16 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.025 (sec). Leaf size: 50

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

\[ y = c_1 \sqrt {x \left (2+x \right )}+\frac {c_2 \left (\left (2+x \right ) \operatorname {arctanh}\left (\frac {\sqrt {2+x}\, \sqrt {2}}{2}\right )-\sqrt {2+x}\, \sqrt {2}\right ) \sqrt {x}}{\sqrt {2+x}} \]

✓ Mathematica. Time used: 0.51 (sec). Leaf size: 92

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

\[ y(x)\to \frac {\exp \left (\int _1^x\frac {5 K[1]+4}{4 K[1]^2+8 K[1]}dK[1]\right ) \left (c_2 \int _1^x\exp \left (-2 \int _1^{K[2]}\frac {5 K[1]+4}{4 K[1]^2+8 K[1]}dK[1]\right )dK[2]+c_1\right )}{\sqrt [4]{2} \sqrt [4]{x+2}} \]

✓ Sympy. Time used: 1.095 (sec). Leaf size: 97

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

\[ y{\left (x \right )} = \sqrt {x} \left (C_{1} + C_{2} \left (\begin {cases} 2 \sqrt {\frac {x}{x + 2} - 1} + 2 \operatorname {asin}{\left (\frac {1}{\sqrt {\frac {x}{x + 2}}} \right )} & \text {for}\: \left |{\frac {x}{x + 2}}\right | > 1 \\2 i \sqrt {- \frac {x}{x + 2} + 1} + i \log {\left (\frac {x}{x + 2} \right )} - 2 i \log {\left (\sqrt {- \frac {x}{x + 2} + 1} + 1 \right )} & \text {otherwise} \end {cases}\right )\right ) \sqrt {x + 2} {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, 0 \\ 1 \end {matrix}\middle | {\frac {x}{x + 2}} \right )} \]