59.1.35 problem 36

Internal problem ID [9207]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 36
Date solved : Wednesday, March 05, 2025 at 07:37:50 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 15

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

\[ y = x \left (\sin \left (x \right ) c_{1} +\cos \left (x \right ) c_{2} \right ) \]

✓ Mathematica. Time used: 0.029 (sec). Leaf size: 33

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

\[ y(x)\to c_1 e^{-i x} x-\frac {1}{2} i c_2 e^{i x} x \]

✓ Sympy. Time used: 0.200 (sec). Leaf size: 20

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

\[ y{\left (x \right )} = x^{\frac {3}{2}} \left (C_{1} J_{\frac {1}{2}}\left (x\right ) + C_{2} Y_{\frac {1}{2}}\left (x\right )\right ) \]