9.4.14 problem problem 14

Internal problem ID [978]
Book : Differential equations and linear algebra, 4th ed., Edwards and Penney
Section : Section 7.3, The eigenvalue method for linear systems. Page 395
Problem number : problem 14
Date solved : Wednesday, February 05, 2025 at 04:51:25 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 45

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

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

✓ Solution by Mathematica

Time used: 0.002 (sec). Leaf size: 51

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

\begin{align*} \text {x1}(t)\to e^{3 t} (c_1 \cos (4 t)-c_2 \sin (4 t)) \\ \text {x2}(t)\to e^{3 t} (c_2 \cos (4 t)+c_1 \sin (4 t)) \\ \end{align*}