67.9.15 problem 14.2 (e)

Internal problem ID [16563]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 14. Higher order equations and the reduction of order method. Additional exercises page 277
Problem number : 14.2 (e)
Date solved : Thursday, October 02, 2025 at 01:36:29 PM
CAS classification : [[_Emden, _Fowler]]

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

Using reduction of order method given that one solution is

\begin{align*} y&=\sqrt {x} \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 14

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

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

✓ Mathematica. Time used: 0.012 (sec). Leaf size: 24

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

\begin{align*} y(x)&\to \frac {1}{2} \sqrt {x} (c_2 \log (x)+2 c_1) \end{align*}

✓ Sympy. Time used: 0.030 (sec). Leaf size: 14

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

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