10.16.12 problem 12

Internal problem ID [1412]
Book : Elementary differential equations and boundary value problems, 10th ed., Boyce and DiPrima
Section : Chapter 7.6, Complex Eigenvalues. page 417
Problem number : 12
Date solved : Monday, January 27, 2025 at 04:56:57 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=-\frac {4 x_{1} \left (t \right )}{5}+2 x_{2} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=-x_{1} \left (t \right )+\frac {6 x_{2} \left (t \right )}{5} \end{align*}

✓ Solution by Maple

Time used: 0.018 (sec). Leaf size: 45

dsolve([diff(x__1(t),t)=-4/5*x__1(t)+2*x__2(t),diff(x__2(t),t)=-1*x__1(t)+6/5*x__2(t)],singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{\frac {t}{5}} \left (c_1 \sin \left (t \right )+c_2 \cos \left (t \right )\right ) \\ x_{2} \left (t \right ) &= \frac {{\mathrm e}^{\frac {t}{5}} \left (\cos \left (t \right ) c_1 +c_2 \cos \left (t \right )+c_1 \sin \left (t \right )-\sin \left (t \right ) c_2 \right )}{2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.005 (sec). Leaf size: 56

DSolve[{D[ x1[t],t]==-4/5*x1[t]+2*x2[t],D[ x2[t],t]==-1*x1[t]+6/5*x2[t]},{x1[t],x2[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to e^{t/5} (c_1 \cos (t)-(c_1-2 c_2) \sin (t)) \\ \text {x2}(t)\to e^{t/5} (c_2 (\sin (t)+\cos (t))-c_1 \sin (t)) \\ \end{align*}