52.3.11 problem 13

Internal problem ID [8297]
Book : DIFFERENTIAL EQUATIONS with Boundary Value Problems. DENNIS G. ZILL, WARREN S. WRIGHT, MICHAEL R. CULLEN. Brooks/Cole. Boston, MA. 2013. 8th edition.
Section : CHAPTER 6 SERIES SOLUTIONS OF LINEAR EQUATIONS. 6.4 SPECIAL FUNCTIONS. EXERCISES 6.4. Page 267
Problem number : 13
Date solved : Wednesday, March 05, 2025 at 05:34:24 AM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} x y^{\prime \prime }+2 y^{\prime }+4 y&=0 \end{align*}

✓ Maple. Time used: 0.026 (sec). Leaf size: 27

ode:=x*diff(diff(y(x),x),x)+2*diff(y(x),x)+4*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {c_{2} \operatorname {BesselY}\left (1, 4 \sqrt {x}\right )+c_{1} \operatorname {BesselJ}\left (1, 4 \sqrt {x}\right )}{\sqrt {x}} \]

✓ Mathematica. Time used: 0.038 (sec). Leaf size: 42

ode=x*D[y[x],{x,2}]+2*D[y[x],x]+4*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {c_1 \operatorname {BesselJ}\left (1,4 \sqrt {x}\right )-2 i c_2 \operatorname {BesselY}\left (1,4 \sqrt {x}\right )}{2 \sqrt {x}} \]

✓ Sympy. Time used: 0.205 (sec). Leaf size: 27

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*Derivative(y(x), (x, 2)) + 4*y(x) + 2*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {C_{1} J_{1}\left (4 \sqrt {x}\right ) + C_{2} Y_{1}\left (4 \sqrt {x}\right )}{\sqrt {x}} \]