59.1.744 problem 764

Internal problem ID [9916]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 764
Date solved : Sunday, March 30, 2025 at 02:49:30 PM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} \left (x^{2}+1\right ) y^{\prime \prime }-6 y&=0 \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 31

ode:=(x^2+1)*diff(diff(y(x),x),x)-6*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {3 x c_2 \left (x^{2}+1\right ) \arctan \left (x \right )}{2}+c_1 \,x^{3}+\frac {3 c_2 \,x^{2}}{2}+c_1 x +c_2 \]

✓ Mathematica. Time used: 0.148 (sec). Leaf size: 33

ode=(x^2+1)*D[y[x],{x,2}]-6*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.334 (sec). Leaf size: 70

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((x**2 + 1)*Derivative(y(x), (x, 2)) - 6*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {\left (x^{2} + 1\right )^{\frac {3}{2}} \left (C_{1} \sqrt {\frac {x^{2}}{x^{2} + 1}} {{}_{2}F_{1}\left (\begin {matrix} -1, 0 \\ \frac {3}{2} \end {matrix}\middle | {\frac {x^{2}}{x^{2} + 1}} \right )} + C_{2} {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{2}, - \frac {1}{2} \\ \frac {1}{2} \end {matrix}\middle | {\frac {x^{2}}{x^{2} + 1}} \right )}\right ) \sqrt [4]{x^{2}}}{\sqrt {x}} \]