10.16.15 problem 25

Internal problem ID [1415]
Book : Elementary differential equations and boundary value problems, 10th ed., Boyce and DiPrima
Section : Chapter 7.6, Complex Eigenvalues. page 417
Problem number : 25
Date solved : Monday, January 27, 2025 at 04:56:59 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=-\frac {x_{1} \left (t \right )}{2}-\frac {x_{2} \left (t \right )}{8}\\ \frac {d}{d t}x_{2} \left (t \right )&=2 x_{1} \left (t \right )-\frac {x_{2} \left (t \right )}{2} \end{align*}

✓ Solution by Maple

Time used: 0.020 (sec). Leaf size: 45

dsolve([diff(x__1(t),t)=-1/2*x__1(t)-1/8*x__2(t),diff(x__2(t),t)=2*x__1(t)-1/2*x__2(t)],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.006 (sec). Leaf size: 68

DSolve[{D[ x1[t],t]==-1/2*x1[t]-1/8*x2[t],D[ x2[t],t]==2*x1[t]-1/2*x2[t]},{x1[t],x2[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to \frac {1}{4} e^{-t/2} \left (4 c_1 \cos \left (\frac {t}{2}\right )-c_2 \sin \left (\frac {t}{2}\right )\right ) \\ \text {x2}(t)\to e^{-t/2} \left (c_2 \cos \left (\frac {t}{2}\right )+4 c_1 \sin \left (\frac {t}{2}\right )\right ) \\ \end{align*}