59.1.316 problem 321

Internal problem ID [9488]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 321
Date solved : Sunday, March 30, 2025 at 02:36:02 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.043 (sec). Leaf size: 45

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

\[ y = \frac {c_2 \left (\sqrt {x^{2}+2}+x \right )^{-\sqrt {2}}+c_1 \left (\sqrt {x^{2}+2}+x \right )^{\sqrt {2}}}{\sqrt {x^{2}+2}} \]

✓ Mathematica. Time used: 0.081 (sec). Leaf size: 92

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

\[ y(x)\to \frac {2^{3/4} c_1 \cos \left (2 \sqrt {2} \arcsin \left (\frac {1}{2} \sqrt {2-i \sqrt {2} x}\right )\right )}{\sqrt {\pi } \sqrt {x^2+2}}+\frac {c_2 Q_{-\frac {1}{2}+\sqrt {2}}^{\frac {1}{2}}\left (\frac {i x}{\sqrt {2}}\right )}{\sqrt [4]{x^2+2}} \]

✗ Sympy

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

False