85.27.10 problem 10

Internal problem ID [22602]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter two. First order and simple higher order ordinary differential equations. A Exercises at page 60
Problem number : 10
Date solved : Thursday, October 02, 2025 at 08:54:54 PM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} x y^{\prime \prime }+2 y&=0 \end{align*}

✓ Maple. Time used: 0.001 (sec). Leaf size: 33

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

\[ y = \sqrt {x}\, \left (\operatorname {BesselY}\left (1, 2 \sqrt {2}\, \sqrt {x}\right ) c_2 +\operatorname {BesselJ}\left (1, 2 \sqrt {2}\, \sqrt {x}\right ) c_1 \right ) \]

✓ Mathematica. Time used: 0.031 (sec). Leaf size: 54

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

\begin{align*} y(x)&\to \sqrt {2} \sqrt {x} \left (c_1 \operatorname {BesselJ}\left (1,2 \sqrt {2} \sqrt {x}\right )+2 i c_2 \operatorname {BesselY}\left (1,2 \sqrt {2} \sqrt {x}\right )\right ) \end{align*}

✓ Sympy. Time used: 0.038 (sec). Leaf size: 37

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

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