76.19.18 problem 18

Internal problem ID [17742]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 5. The Laplace transform. Section 5.4 (Solving differential equations with Laplace transform). Problems at page 327
Problem number : 18
Date solved : Tuesday, January 28, 2025 at 11:02:05 AM
CAS classification : system_of_ODEs

\begin{align*} y_{1}^{\prime }\left (t \right )&=-4 y_{1} \left (t \right )-y_{2} \left (t \right )\\ y_{2}^{\prime }\left (t \right )&=y_{1} \left (t \right )-2 y_{2} \left (t \right ) \end{align*}

With initial conditions

\begin{align*} y_{1} \left (0\right ) = 1\\ y_{2} \left (0\right ) = 0 \end{align*}

Solution by Maple

Time used: 0.051 (sec). Leaf size: 23

dsolve([diff(y__1(t),t) = -4*y__1(t)-y__2(t), diff(y__2(t),t) = y__1(t)-2*y__2(t), y__1(0) = 1, y__2(0) = 0], singsol=all)
 
\begin{align*} y_{1} \left (t \right ) &= {\mathrm e}^{-3 t} \left (1-t \right ) \\ y_{2} \left (t \right ) &= t \,{\mathrm e}^{-3 t} \\ \end{align*}

Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 25

DSolve[{D[y1[t],t]==-4*y1[t]-y2[t],D[y2[t],t]==1*y1[t]-2*y2[t]},{y1[0]==1,y2[0]==0},{y1[t],y2[t]},t,IncludeSingularSolutions -> True]
 
\begin{align*} \text {y1}(t)\to -e^{-3 t} (t-1) \\ \text {y2}(t)\to e^{-3 t} t \\ \end{align*}