76.19.19 problem 19

Internal problem ID [17743]
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 : 19
Date solved : Tuesday, January 28, 2025 at 11:02:05 AM
CAS classification : system_of_ODEs

\begin{align*} y_{1}^{\prime }\left (t \right )&=2 y_{1} \left (t \right )-64 y_{2} \left (t \right )\\ y_{2}^{\prime }\left (t \right )&=y_{1} \left (t \right )-14 y_{2} \left (t \right ) \end{align*}

With initial conditions

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

✓ Solution by Maple

Time used: 0.056 (sec). Leaf size: 25

dsolve([diff(y__1(t),t) = 2*y__1(t)-64*y__2(t), diff(y__2(t),t) = y__1(t)-14*y__2(t), y__1(0) = 0, y__2(0) = 1], singsol=all)
 

\begin{align*} y_{1} \left (t \right ) &= -64 \,{\mathrm e}^{-6 t} t \\ y_{2} \left (t \right ) &= \frac {{\mathrm e}^{-6 t} \left (-512 t +64\right )}{64} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 27

DSolve[{D[y1[t],t]==2*y1[t]-64*y2[t],D[y2[t],t]==1*y1[t]-14*y2[t]},{y1[0]==0,y2[0]==1},{y1[t],y2[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {y1}(t)\to -64 e^{-6 t} t \\ \text {y2}(t)\to e^{-6 t} (1-8 t) \\ \end{align*}