76.19.24 problem 24

Internal problem ID [17748]
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 : 24
Date solved : Tuesday, January 28, 2025 at 11:02:09 AM
CAS classification : system_of_ODEs

\begin{align*} y_{1}^{\prime }\left (t \right )&=y_{2} \left (t \right )-y_{3} \left (t \right )\\ y_{2}^{\prime }\left (t \right )&=y_{1} \left (t \right )+y_{3} \left (t \right )-{\mathrm e}^{-t}\\ y_{3}^{\prime }\left (t \right )&=y_{1} \left (t \right )+y_{2} \left (t \right )+{\mathrm e}^{t} \end{align*}

With initial conditions

\begin{align*} y_{1} \left (0\right ) = 1\\ y_{2} \left (0\right ) = 2\\ y_{3} \left (0\right ) = 3 \end{align*}

✓ Solution by Maple

Time used: 0.337 (sec). Leaf size: 56

dsolve([diff(y__1(t),t) = y__2(t)-y__3(t), diff(y__2(t),t) = y__1(t)+y__3(t)-exp(-t), diff(y__3(t),t) = y__1(t)+y__2(t)+exp(t), y__1(0) = 1, y__2(0) = 2, y__3(0) = 3], singsol=all)
 

\begin{align*} y_{1} \left (t \right ) &= {\mathrm e}^{-t} t +\frac {3 \,{\mathrm e}^{-t}}{2}-\frac {{\mathrm e}^{t}}{2} \\ y_{2} \left (t \right ) &= -{\mathrm e}^{-t} t -{\mathrm e}^{-t}+3 \,{\mathrm e}^{t} \\ y_{3} \left (t \right ) &= -\frac {{\mathrm e}^{-t}}{2}+\frac {7 \,{\mathrm e}^{t}}{2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.150 (sec). Leaf size: 108

DSolve[{D[y1[t],t]==y2[t]-y3[t],D[y2[t],t]==y1[t]+y3[t]-Exp[-t],D[y3[t],t]==y1[t]+y2[t]+Exp[t]},{y1[0]==1,y2[0]==2,y3[0]==3},{y1[t],y2[t],y3[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {y1}(t)\to \log \left (e^t\right )-\frac {1}{2} e^{-t} \left (2 e^t t-2 t+e^{2 t}-3\right ) \\ \text {y2}(t)\to -e^t (t-3)-e^{-t}-e^{-t} t+t+\left (e^t-1\right ) \log \left (e^t\right ) \\ \text {y3}(t)\to e^t \left (\frac {7}{2}-t\right )-\frac {e^{-t}}{2}+t+\left (e^t-1\right ) \log \left (e^t\right ) \\ \end{align*}