53.1.524 problem 540

Internal problem ID [10996]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 540
Date solved : Tuesday, September 30, 2025 at 07:34:54 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 2 x^{2} \left (x^{2}+2\right ) y^{\prime \prime }-x \left (-7 x^{2}+12\right ) y^{\prime }+\left (3 x^{2}+7\right ) y&=0 \end{align*}

✓ Maple. Time used: 0.050 (sec). Leaf size: 35

ode:=2*x^2*(x^2+2)*diff(diff(y(x),x),x)-x*(-7*x^2+12)*diff(y(x),x)+(3*x^2+7)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {c_1 \,x^{{7}/{2}}}{\left (2 x^{2}+4\right )^{{9}/{4}}}+c_2 \sqrt {x}\, \operatorname {hypergeom}\left (\left [\frac {3}{4}, 1\right ], \left [-\frac {1}{2}\right ], -\frac {x^{2}}{2}\right ) \]

✓ Mathematica. Time used: 0.182 (sec). Leaf size: 117

ode=2*x^2*(2+x^2)*D[y[x],{x,2}]-x*(12-7*x^2)*D[y[x],x]+(7+3*x^2)*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \exp \left (\int _1^x\frac {3 K[1]^2+16}{4 K[1]^3+8 K[1]}dK[1]-\frac {1}{2} \int _1^x\left (\frac {13 K[2]}{2 \left (K[2]^2+2\right )}-\frac {3}{K[2]}\right )dK[2]\right ) \left (c_2 \int _1^x\exp \left (-2 \int _1^{K[3]}\frac {3 K[1]^2+16}{4 K[1]^3+8 K[1]}dK[1]\right )dK[3]+c_1\right ) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*x**2*(x**2 + 2)*Derivative(y(x), (x, 2)) - x*(12 - 7*x**2)*Derivative(y(x), x) + (3*x**2 + 7)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False