76.19.15 problem 15

Internal problem ID [17739]
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 : 15
Date solved : Tuesday, January 28, 2025 at 11:02:02 AM
CAS classification : system_of_ODEs

\begin{align*} y_{1}^{\prime }\left (t \right )&=5 y_{1} \left (t \right )-2 y_{2} \left (t \right )\\ y_{2}^{\prime }\left (t \right )&=6 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: 0.191 (sec). Leaf size: 29

dsolve([diff(y__1(t),t) = 5*y__1(t)-2*y__2(t), diff(y__2(t),t) = 6*y__1(t)-2*y__2(t), y__1(0) = 1, y__2(0) = 0], singsol=all)
 

\begin{align*} y_{1} \left (t \right ) &= 4 \,{\mathrm e}^{2 t}-3 \,{\mathrm e}^{t} \\ y_{2} \left (t \right ) &= 6 \,{\mathrm e}^{2 t}-6 \,{\mathrm e}^{t} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 29

DSolve[{D[y1[t],t]==5*y1[t]-2*y2[t],D[y2[t],t]==6*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^t \left (4 e^t-3\right ) \\ \text {y2}(t)\to 6 e^t \left (e^t-1\right ) \\ \end{align*}