53.1.300 problem 303

Internal problem ID [10772]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 303
Date solved : Tuesday, September 30, 2025 at 07:31:49 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 17

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

\[ y = c_1 \left (-1+x \right )+c_2 \,{\mathrm e}^{x} x^{2} \]

✓ Mathematica. Time used: 0.228 (sec). Leaf size: 126

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

\begin{align*} y(x)&\to \exp \left (\int _1^x\left (-\frac {K[1]}{K[1]^2-2}+\frac {1}{2}+\frac {3}{2 K[1]}\right )dK[1]-\frac {1}{2} \int _1^x\left (-\frac {2 K[2]}{K[2]^2-2}-1-\frac {1}{K[2]}\right )dK[2]\right ) \left (c_2 \int _1^x\exp \left (-2 \int _1^{K[3]}\frac {K[1]^3+K[1]^2-2 K[1]-6}{2 K[1] \left (K[1]^2-2\right )}dK[1]\right )dK[3]+c_1\right ) \end{align*}

✗ Sympy

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

False