10.16.1 problem 1

Internal problem ID [1401]
Book : Elementary differential equations and boundary value problems, 10th ed., Boyce and DiPrima
Section : Chapter 7.6, Complex Eigenvalues. page 417
Problem number : 1
Date solved : Monday, January 27, 2025 at 04:56:47 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.016 (sec). Leaf size: 52

dsolve([diff(x__1(t),t)=3*x__1(t)-2*x__2(t),diff(x__2(t),t)=4*x__1(t)-1*x__2(t)],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.005 (sec). Leaf size: 58

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

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