23.5.4 problem 3(c)

Internal problem ID [4181]
Book : Theory and solutions of Ordinary Differential equations, Donald Greenspan, 1960
Section : Chapter 7. Special functions. Exercises at page 124
Problem number : 3(c)
Date solved : Sunday, March 30, 2025 at 02:41:47 AM
CAS classification : [_Lienard]

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

Using series method with expansion around

\begin{align*} 0 \end{align*}

✓ Maple. Time used: 0.019 (sec). Leaf size: 32

Order:=6; 
ode:=diff(diff(y(x),x),x)+2/x*diff(y(x),x)+y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = c_1 \left (1-\frac {1}{6} x^{2}+\frac {1}{120} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_2 \left (1-\frac {1}{2} x^{2}+\frac {1}{24} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x} \]

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 43

ode=D[y[x],{x,2}]+2/x*D[y[x],x]+y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},{y[x]},{x,0,5}]
 

\[ \{y(x)\}\to c_1 \left (\frac {x^3}{24}-\frac {x}{2}+\frac {1}{x}\right )+c_2 \left (\frac {x^4}{120}-\frac {x^2}{6}+1\right ) \]

✓ Sympy. Time used: 0.759 (sec). Leaf size: 39

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(y(x) + Derivative(y(x), (x, 2)) + 2*Derivative(y(x), x)/x,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (\frac {x^{4}}{120} - \frac {x^{2}}{6} + 1\right ) + \frac {C_{1} \left (- \frac {x^{6}}{720} + \frac {x^{4}}{24} - \frac {x^{2}}{2} + 1\right )}{x} + O\left (x^{6}\right ) \]