44.12.6 problem 5

Internal problem ID [9296]
Book : Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section : Chapter 2. Second-Order Linear Equations. Section 2.4. THE USE OF A KNOWN SOLUTION TO FIND ANOTHER. Page 74
Problem number : 5
Date solved : Tuesday, September 30, 2025 at 06:16:14 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {1}{4}\right ) y&=0 \end{align*}

Using reduction of order method given that one solution is

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

✓ Maple. Time used: 0.015 (sec). Leaf size: 17

ode:=x^2*diff(diff(y(x),x),x)+x*diff(y(x),x)+(x^2-1/4)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.021 (sec). Leaf size: 39

ode=x^2*D[y[x],{x,2}]+x*D[y[x],x]+(x^2-1/4)*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {e^{-i x} \left (2 c_1-i c_2 e^{2 i x}\right )}{2 \sqrt {x}} \end{align*}

✓ Sympy. Time used: 0.136 (sec). Leaf size: 15

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) + x*Derivative(y(x), x) + (x**2 - 1/4)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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