50.12.1 problem 1(a)

Internal problem ID [8005]
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 : 1(a)
Date solved : Wednesday, March 05, 2025 at 05:23:14 AM
CAS classification : [[_2nd_order, _missing_x]]

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

Using reduction of order method given that one solution is

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

✓ Maple. Time used: 0.000 (sec). Leaf size: 13

ode:=diff(diff(y(x),x),x)+y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \sin \left (x \right ) c_{1} +\cos \left (x \right ) c_{2} \]

✓ Mathematica. Time used: 0.011 (sec). Leaf size: 16

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

\[ y(x)\to c_1 \cos (x)+c_2 \sin (x) \]

✓ Sympy. Time used: 0.048 (sec). Leaf size: 12

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

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