78.3.24 problem 7.b

Internal problem ID [21020]
Book : A FIRST COURSE IN DIFFERENTIAL EQUATIONS FOR SCIENTISTS AND ENGINEERS. By Russell Herman. University of North Carolina Wilmington. LibreText. compiled on 06/09/2025
Section : Chapter 4, Series solutions. Problems section 4.9
Problem number : 7.b
Date solved : Thursday, October 02, 2025 at 07:01:30 PM
CAS classification : [[_Emden, _Fowler]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.016 (sec). Leaf size: 19

Order:=6; 
ode:=diff(diff(y(x),x),x)+1/4/x^2*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \sqrt {x}\, \left (c_1 +c_2 \ln \left (x \right )\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 22

ode=D[y[x],{x,2}]+1/(4*x^2)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 \sqrt {x}+c_2 \sqrt {x} \log (x) \]

✓ Sympy. Time used: 0.148 (sec). Leaf size: 12

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(Derivative(y(x), (x, 2)) + y(x)/(4*x**2),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{1} \sqrt {x} + O\left (x^{6}\right ) \]