68.1.2 problem Problem 1.1(b)

Internal problem ID [14052]
Book : Differential Equations, Linear, Nonlinear, Ordinary, Partial. A.C. King, J.Billingham, S.R.Otto. Cambridge Univ. Press 2003
Section : Chapter 1 VARIABLE COEFFICIENT, SECOND ORDER DIFFERENTIAL EQUATIONS. Problems page 28
Problem number : Problem 1.1(b)
Date solved : Wednesday, March 05, 2025 at 10:27:37 PM
CAS classification : [_Lienard]

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

Using reduction of order method given that one solution is

\begin{align*} y&=\frac {\sin \left (x \right )}{x} \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 17

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

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

✓ Mathematica. Time used: 0.042 (sec). Leaf size: 37

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

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

✓ Sympy. Time used: 0.205 (sec). Leaf size: 20

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

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