81.10.1 problem 14-1

Internal problem ID [21596]
Book : The Differential Equations Problem Solver. VOL. I. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 14. Second order homogeneous differential equations with constant coefficients. Page 297.
Problem number : 14-1
Date solved : Thursday, October 02, 2025 at 07:58:50 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} x^{\prime \prime }-4 x&=0 \end{align*}

With initial conditions

\begin{align*} x \left (0\right )&=0 \\ x^{\prime }\left (0\right )&=3 \\ \end{align*}

✓ Maple. Time used: 0.017 (sec). Leaf size: 10

ode:=diff(diff(x(t),t),t)-4*x(t) = 0; 
ic:=[x(0) = 0, D(x)(0) = 3]; 
dsolve([ode,op(ic)],x(t), singsol=all);
 

\[ x = \frac {3 \sinh \left (2 t \right )}{2} \]

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 21

ode=D[x[t],{t,2}]-4*x[t]==0; 
ic={x[0]==0,Derivative[1][x][0] ==3}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {3}{4} e^{-2 t} \left (e^{4 t}-1\right ) \end{align*}

✓ Sympy. Time used: 0.048 (sec). Leaf size: 19

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

\[ x{\left (t \right )} = \frac {3 e^{2 t}}{4} - \frac {3 e^{- 2 t}}{4} \]