74.20.3 problem 3

Internal problem ID [16635]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 5. Applications of Higher Order Equations. Exercises 5.2, page 241
Problem number : 3
Date solved : Tuesday, January 28, 2025 at 09:13:40 AM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} \frac {x^{\prime \prime }}{4}+2 x^{\prime }+x&=0 \end{align*}

With initial conditions

\begin{align*} x \left (0\right )&=-{\frac {1}{2}}\\ x^{\prime }\left (0\right )&=1 \end{align*}

✓ Solution by Maple

Time used: 0.094 (sec). Leaf size: 39

dsolve([1/4*diff(x(t),t$2)+2*diff(x(t),t)+x(t)=0,x(0) = -1/2, D(x)(0) = 1],x(t), singsol=all)
 

\[ x \left (t \right ) = \frac {\left (-3-\sqrt {3}\right ) {\mathrm e}^{2 \left (-2+\sqrt {3}\right ) t}}{12}+\frac {{\mathrm e}^{-2 \left (2+\sqrt {3}\right ) t} \left (-3+\sqrt {3}\right )}{12} \]

✓ Solution by Mathematica

Time used: 0.024 (sec). Leaf size: 47

DSolve[{1/4*D[x[t],{t,2}]+2*D[x[t],t]+x[t]==0,{x[0]==-1/2,Derivative[1][x][0 ]==1}},x[t],t,IncludeSingularSolutions -> True]
 

\[ x(t)\to \frac {1}{12} e^{-2 \left (2+\sqrt {3}\right ) t} \left (-\left (3+\sqrt {3}\right ) e^{4 \sqrt {3} t}-3+\sqrt {3}\right ) \]