10.16.13 problem 23

Internal problem ID [1413]
Book : Elementary differential equations and boundary value problems, 10th ed., Boyce and DiPrima
Section : Chapter 7.6, Complex Eigenvalues. page 417
Problem number : 23
Date solved : Monday, January 27, 2025 at 04:56:58 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=-\frac {x_{1} \left (t \right )}{4}+x_{2} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=-x_{1} \left (t \right )-\frac {x_{2} \left (t \right )}{4}\\ \frac {d}{d t}x_{3} \left (t \right )&=-\frac {x_{3} \left (t \right )}{4} \end{align*}

✓ Solution by Maple

Time used: 0.022 (sec). Leaf size: 45

dsolve([diff(x__1(t),t)=-1/4*x__1(t)+1*x__2(t)+0*x__3(t),diff(x__2(t),t)=-1*x__1(t)-1/4*x__2(t)+0*x__3(t),diff(x__3(t),t)=0*x__1(t)-0*x__2(t)-1/4*x__3(t)],singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{-\frac {t}{4}} \left (c_1 \sin \left (t \right )+c_2 \cos \left (t \right )\right ) \\ x_{2} \left (t \right ) &= {\mathrm e}^{-\frac {t}{4}} \left (\cos \left (t \right ) c_1 -\sin \left (t \right ) c_2 \right ) \\ x_{3} \left (t \right ) &= c_3 \,{\mathrm e}^{-\frac {t}{4}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.021 (sec). Leaf size: 110

DSolve[{D[ x1[t],t]==-1/4*x1[t]+1*x2[t]+0*x3[t],D[ x2[t],t]==-1*x1[t]-1/4*x2[t]+0*x3[t],D[ x3[t],t]==0*x1[t]-0*x2[t]-1/4*x3[t]},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to e^{-t/4} (c_1 \cos (t)+c_2 \sin (t)) \\ \text {x2}(t)\to e^{-t/4} (c_2 \cos (t)-c_1 \sin (t)) \\ \text {x3}(t)\to c_3 e^{-t/4} \\ \text {x1}(t)\to e^{-t/4} (c_1 \cos (t)+c_2 \sin (t)) \\ \text {x2}(t)\to e^{-t/4} (c_2 \cos (t)-c_1 \sin (t)) \\ \text {x3}(t)\to 0 \\ \end{align*}