59.1.112 problem 114

Internal problem ID [9284]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 114
Date solved : Sunday, March 30, 2025 at 02:31:29 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 4 x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+4 x \left (6 x^{2}+1\right ) y^{\prime }-\left (-25 x^{2}+1\right ) y&=0 \end{align*}

✓ Maple. Time used: 0.032 (sec). Leaf size: 34

ode:=4*x^2*(x^2+1)*diff(diff(y(x),x),x)+4*x*(6*x^2+1)*diff(y(x),x)-(-25*x^2+1)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.454 (sec). Leaf size: 70

ode=4*x^2*(1+x^2)*D[y[x],{x,2}]+4*x*(1+6*x^2)*D[y[x],x]-(1-25*x^2)*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {\left (c_2 x \text {arcsinh}(x)-c_2 \sqrt {x^2+1}+c_1 x\right ) \exp \left (-\frac {1}{2} \int _1^x\frac {6 K[1]^2+1}{K[1]^3+K[1]}dK[1]\right )}{\sqrt [4]{x^2+1}} \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*x**2*(x**2 + 1)*Derivative(y(x), (x, 2)) + 4*x*(6*x**2 + 1)*Derivative(y(x), x) - (1 - 25*x**2)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False