76.6.1 problem 1

Internal problem ID [17456]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 3. Systems of two first order equations. Section 3.2 (Two first order linear differential equations). Problems at page 142
Problem number : 1
Date solved : Tuesday, January 28, 2025 at 10:38:59 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=x \left (t \right )+4 \end{align*}

✓ Solution by Maple

Time used: 0.062 (sec). Leaf size: 31

dsolve([diff(x(t),t)=y(t),diff(y(t),t)=x(t)+4],singsol=all)
 

\begin{align*} x \left (t \right ) &= c_{2} {\mathrm e}^{-t}+c_{1} {\mathrm e}^{t}-4 \\ y \left (t \right ) &= -c_{2} {\mathrm e}^{-t}+c_{1} {\mathrm e}^{t} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.041 (sec). Leaf size: 70

DSolve[{D[x[t],t]==y[t],D[y[t],t]==x[t]+4},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to \frac {1}{2} e^{-t} \left (-8 e^t+(c_1+c_2) e^{2 t}+c_1-c_2\right ) \\ y(t)\to \frac {1}{2} e^{-t} \left (c_1 \left (e^{2 t}-1\right )+c_2 \left (e^{2 t}+1\right )\right ) \\ \end{align*}