59.1.610 problem 626

Internal problem ID [9782]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 626
Date solved : Sunday, March 30, 2025 at 02:46:31 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+x \left (2 x^{2}+5\right ) y^{\prime }-21 y&=0 \end{align*}

✓ Maple. Time used: 0.018 (sec). Leaf size: 41

ode:=x^2*(x^2+1)*diff(diff(y(x),x),x)+x*(2*x^2+5)*diff(y(x),x)-21*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {c_1 \left (x^{2}+1\right )^{{5}/{2}} \left (x^{2}+8\right )+35 \left (x^{6}+4 x^{4}+\frac {24}{5} x^{2}+\frac {64}{35}\right ) c_2}{x^{7}} \]

✓ Mathematica. Time used: 0.431 (sec). Leaf size: 126

ode=x^2*(1+x^2)*D[y[x],{x,2}]+x*(5+2*x^2)*D[y[x],x]-21*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \left (x^2+8\right ) \exp \left (\int _1^x-\frac {2 K[1]^2+9}{2 \left (K[1]^3+K[1]\right )}dK[1]-\frac {1}{2} \int _1^x\frac {2 K[2]^2+5}{K[2]^3+K[2]}dK[2]\right ) \left (c_2 \int _1^x\frac {\exp \left (-2 \int _1^{K[3]}-\frac {2 K[1]^2+9}{2 \left (K[1]^3+K[1]\right )}dK[1]\right )}{\left (K[3]^2+8\right )^2}dK[3]+c_1\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*(x**2 + 1)*Derivative(y(x), (x, 2)) + x*(2*x**2 + 5)*Derivative(y(x), x) - 21*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False