10.19.4 problem 4

Internal problem ID [1445]
Book : Elementary differential equations and boundary value problems, 10th ed., Boyce and DiPrima
Section : Chapter 9.1, The Phase Plane: Linear Systems. page 505
Problem number : 4
Date solved : Monday, January 27, 2025 at 04:57:26 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=x_{1} \left (t \right )-4 x_{2} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=4 x_{1} \left (t \right )-7 x_{2} \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 34

dsolve([diff(x__1(t),t)=1*x__1(t)-4*x__2(t),diff(x__2(t),t)=4*x__1(t)-7*x__2(t)],singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{-3 t} \left (c_2 t +c_1 \right ) \\ x_{2} \left (t \right ) &= \frac {{\mathrm e}^{-3 t} \left (4 c_2 t +4 c_1 -c_2 \right )}{4} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.002 (sec). Leaf size: 46

DSolve[{D[ x1[t],t]==1*x1[t]-4*x2[t],D[ x2[t],t]==4*x1[t]-7*x2[t]},{x1[t],x2[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to e^{-3 t} (4 c_1 t-4 c_2 t+c_1) \\ \text {x2}(t)\to e^{-3 t} (4 (c_1-c_2) t+c_2) \\ \end{align*}