59.1.525 problem 541

Internal problem ID [9697]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 541
Date solved : Wednesday, March 05, 2025 at 07:57:37 AM
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}+4\right ) y^{\prime }-\left (-3 x^{2}+1\right ) y&=0 \end{align*}

✓ Maple. Time used: 0.057 (sec). Leaf size: 35

ode:=2*x^2*(x^2+2)*diff(diff(y(x),x),x)+x*(7*x^2+4)*diff(y(x),x)-(-3*x^2+1)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {c_{2} \operatorname {LegendreQ}\left (-\frac {1}{4}, \frac {1}{4}, \frac {i \sqrt {2}\, x}{2}\right ) \left (x^{2}+2\right )^{{1}/{8}}+c_{1}}{\left (x^{2}+2\right )^{{1}/{4}} \sqrt {x}} \]

✓ Mathematica. Time used: 0.257 (sec). Leaf size: 95

ode=2*x^2*(2+x^2)*D[y[x],{x,2}]+x*(4+7*x^2)*D[y[x],x]-(1-3*x^2)*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {\left (c_2 \sqrt [8]{x^2+2} \operatorname {Gamma}\left (\frac {3}{4}\right ) Q_{-\frac {1}{4}}^{\frac {1}{4}}\left (\frac {i x}{\sqrt {2}}\right )+2^{3/8} c_1\right ) \exp \left (\int _1^x-\frac {3 K[1]^2+4}{4 K[1]^3+8 K[1]}dK[1]\right )}{\sqrt [8]{x^2+2} \operatorname {Gamma}\left (\frac {3}{4}\right )} \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*x**2*(x**2 + 2)*Derivative(y(x), (x, 2)) + x*(7*x**2 + 4)*Derivative(y(x), x) - (1 - 3*x**2)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False