7.25.6 problem 6

Internal problem ID [626]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 5. Linear systems of differential equations. Section 5.4 (The eigenvalue method for homogeneous systems). Problems at page 378
Problem number : 6
Date solved : Monday, January 27, 2025 at 02:56:29 AM
CAS classification : system_of_ODEs

\begin{align*} x_{1}^{\prime }\left (t \right )&=9 x_{1} \left (t \right )+5 x_{2} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=-6 x_{1} \left (t \right )-2 x_{2} \left (t \right ) \end{align*}

With initial conditions

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

✓ Solution by Maple

Time used: 0.020 (sec). Leaf size: 33

dsolve([diff(x__1(t),t) = 9*x__1(t)+5*x__2(t), diff(x__2(t),t) = -6*x__1(t)-2*x__2(t), x__1(0) = 1, x__2(0) = 0], singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= 6 \,{\mathrm e}^{4 t}-5 \,{\mathrm e}^{3 t} \\ x_{2} \left (t \right ) &= -6 \,{\mathrm e}^{4 t}+6 \,{\mathrm e}^{3 t} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 33

DSolve[{D[x1[t],t]==9*x1[t]+5*x2[t],D[x2[t],t]==-6*x1[t]-2*x2[t]},{x1[0]==1,x2[0]==0},{x1[t],x2[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to e^{3 t} \left (6 e^t-5\right ) \\ \text {x2}(t)\to -6 e^{3 t} \left (e^t-1\right ) \\ \end{align*}