70.13.36 problem 36

Internal problem ID [18905]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 4. Second order linear equations. Section 4.3 (Linear homogeneous equations with constant coefficients). Problems at page 239
Problem number : 36
Date solved : Thursday, October 02, 2025 at 03:32:41 PM
CAS classification : [[_2nd_order, _missing_x]]

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

With initial conditions

\begin{align*} y \left (\frac {\pi }{3}\right )&=2 \\ y^{\prime }\left (\frac {\pi }{3}\right )&=-4 \\ \end{align*}

✓ Maple. Time used: 0.085 (sec). Leaf size: 22

ode:=diff(diff(y(x),x),x)+y(x) = 0; 
ic:=[y(1/3*Pi) = 2, D(y)(1/3*Pi) = -4]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \left (\sin \left (x \right )+2 \cos \left (x \right )\right ) \sqrt {3}+\cos \left (x \right )-2 \sin \left (x \right ) \]

✓ Mathematica. Time used: 0.008 (sec). Leaf size: 28

ode=D[y[x],{x,2}]+y[x]==0; 
ic={y[Pi/3]==2,Derivative[1][y][Pi/3] ==-4}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \left (\sqrt {3}-2\right ) \sin (x)+\left (1+2 \sqrt {3}\right ) \cos (x) \end{align*}

✓ Sympy. Time used: 0.040 (sec). Leaf size: 24

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(y(x) + Derivative(y(x), (x, 2)),0) 
ics = {y(pi/3): 2, Subs(Derivative(y(x), x), x, pi/3): -4} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (-2 + \sqrt {3}\right ) \sin {\left (x \right )} + \left (1 + 2 \sqrt {3}\right ) \cos {\left (x \right )} \]