86.6.3 problem 3

Internal problem ID [23136]
Book : An introduction to Differential Equations. By Howard Frederick Cleaves. 1969. Oliver and Boyd publisher. ISBN 0050015044
Section : Chapter 5. Linear equations of the second order with constant coefficients. Exercise 5b at page 77
Problem number : 3
Date solved : Thursday, October 02, 2025 at 09:23:19 PM
CAS classification : [[_2nd_order, _missing_x]]

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

With initial conditions

\begin{align*} z \left (\frac {\pi }{3 \sqrt {g}}\right )&=5 \\ z \left (\frac {2 \pi }{3 \sqrt {g}}\right )&=\frac {\pi }{3} \\ \end{align*}

✓ Maple. Time used: 0.039 (sec). Leaf size: 31

ode:=diff(diff(z(t),t),t)+g*z(t) = 0; 
ic:=[z(1/3*Pi/g^(1/2)) = 5, z(2/3*Pi/g^(1/2)) = 1/3*Pi]; 
dsolve([ode,op(ic)],z(t), singsol=all);
 

\[ z = \frac {\left (\pi +15\right ) \sqrt {3}\, \sin \left (\sqrt {g}\, t \right )}{9}+\left (-\frac {\pi }{3}+5\right ) \cos \left (\sqrt {g}\, t \right ) \]

✓ Mathematica. Time used: 0.011 (sec). Leaf size: 40

ode=D[z[t],{t,2}]+g*z[t]==0; 
ic={z[Pi/(3*Sqrt[g]) ]==5,z[2*Pi/(3*Sqrt[g])]==Pi/3}; 
DSolve[{ode,ic},z[t],t,IncludeSingularSolutions->True]
 

\begin{align*} z(t)&\to \frac {1}{9} \left (\sqrt {3} (15+\pi ) \sin \left (\sqrt {g} t\right )-3 (\pi -15) \cos \left (\sqrt {g} t\right )\right ) \end{align*}

✓ Sympy. Time used: 0.088 (sec). Leaf size: 138

from sympy import * 
t = symbols("t") 
g = symbols("g") 
z = Function("z") 
ode = Eq(g*z(t) + Derivative(z(t), (t, 2)),0) 
ics = {z(pi/(3*sqrt(g))): 5, z(2*pi/(3*sqrt(g))): pi/3} 
dsolve(ode,func=z(t),ics=ics)
 

\[ z{\left (t \right )} = \frac {\left (\pi e^{\frac {\pi \sqrt {- g}}{3 \sqrt {g}}} - 15\right ) e^{t \sqrt {- g}}}{- 3 e^{\frac {\pi \sqrt {- g}}{3 \sqrt {g}}} + 3 e^{\frac {\pi \sqrt {- g}}{\sqrt {g}}}} + \frac {\left (- \pi e^{\frac {2 \pi \sqrt {- g}}{3 \sqrt {g}}} + 15 e^{\frac {\pi \sqrt {- g}}{\sqrt {g}}}\right ) e^{- t \sqrt {- g}}}{3 e^{\frac {2 \pi \sqrt {- g}}{3 \sqrt {g}}} - 3} \]