59.1.38 problem 40

Internal problem ID [9210]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 40
Date solved : Wednesday, March 05, 2025 at 07:37:52 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: 0.006 (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);
 

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

✓ Mathematica. Time used: 0.028 (sec). Leaf size: 25

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]
 

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

✓ Sympy. Time used: 0.210 (sec). Leaf size: 24

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)
 

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