10.17.7 problem 7

Internal problem ID [1422]
Book : Elementary differential equations and boundary value problems, 10th ed., Boyce and DiPrima
Section : Chapter 7.8, Repeated Eigenvalues. page 436
Problem number : 7
Date solved : Monday, January 27, 2025 at 04:57:05 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*}

With initial conditions

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

✓ Solution by Maple

Time used: 0.016 (sec). Leaf size: 28

dsolve([diff(x__1(t),t) = x__1(t)-4*x__2(t), diff(x__2(t),t) = 4*x__1(t)-7*x__2(t), x__1(0) = 3, x__2(0) = 2], singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{-3 t} \left (4 t +3\right ) \\ x_{2} \left (t \right ) &= \frac {{\mathrm e}^{-3 t} \left (16 t +8\right )}{4} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 34

DSolve[{D[ x1[t],t]==1*x1[t]-4*x2[t],D[ x2[t],t]==1*x1[t]-4*x2[t]},{x1[0]==3,x2[0]==2},{x1[t],x2[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to \frac {5 e^{-3 t}}{3}+\frac {4}{3} \\ \text {x2}(t)\to \frac {5 e^{-3 t}}{3}+\frac {1}{3} \\ \end{align*}