47.4.12 problem 13

Internal problem ID [9786]
Book : Elementary differential equations. By Earl D. Rainville, Phillip E. Bedient. Macmilliam Publishing Co. NY. 6th edition. 1981.
Section : CHAPTER 16. Nonlinear equations. Section 101. Independent variable missing. EXERCISES Page 324
Problem number : 13
Date solved : Tuesday, September 30, 2025 at 06:41:14 PM
CAS classification : [[_2nd_order, _missing_x]]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 17

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

\[ y = c_1 \sin \left (\beta x \right )+c_2 \cos \left (\beta x \right ) \]

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 20

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

\begin{align*} y(x)&\to c_1 \cos (\beta x)+c_2 \sin (\beta x) \end{align*}

✓ Sympy. Time used: 0.044 (sec). Leaf size: 19

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

\[ y{\left (x \right )} = C_{1} e^{- i \beta x} + C_{2} e^{i \beta x} \]