7.17.1 problem 1

Internal problem ID [514]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 3. Power series methods. Section 3.6 (Applications of Bessel functions). Problems at page 261
Problem number : 1
Date solved : Saturday, March 29, 2025 at 04:55:34 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 17

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

\[ y = x \left (\operatorname {BesselY}\left (0, x\right ) c_2 +\operatorname {BesselJ}\left (0, x\right ) c_1 \right ) \]

✓ Mathematica. Time used: 0.019 (sec). Leaf size: 20

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

\[ y(x)\to x (c_1 \operatorname {BesselJ}(0,x)+c_2 \operatorname {BesselY}(0,x)) \]

✓ Sympy. Time used: 0.204 (sec). Leaf size: 14

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

\[ y{\left (x \right )} = x \left (C_{1} J_{0}\left (x\right ) + C_{2} Y_{0}\left (x\right )\right ) \]