15.18.39 problem 39

Internal problem ID [3282]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 35, page 157
Problem number : 39
Date solved : Monday, January 27, 2025 at 07:30:47 AM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} x^{\prime \prime }-k^{2} x&=0 \end{align*}

With initial conditions

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

✓ Solution by Maple

Time used: 0.018 (sec). Leaf size: 22

dsolve([diff(x(t),t$2)-k^2*x(t)=0,x(0) = 0, D(x)(0) = v__0],x(t), singsol=all)
 

\[ x = \frac {v_{0} \left (-{\mathrm e}^{-k t}+{\mathrm e}^{k t}\right )}{2 k} \]

✓ Solution by Mathematica

Time used: 0.016 (sec). Leaf size: 27

DSolve[{D[x[t],{t,2}]-k^2*x[t]==0,{x[0]==0,Derivative[1][x][0 ]==v0}},x[t],t,IncludeSingularSolutions -> True]
 

\[ x(t)\to \frac {\text {v0} e^{-k t} \left (e^{2 k t}-1\right )}{2 k} \]