14.23.6 problem 6

Internal problem ID [2745]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Section 3.9, Systems of differential equations. Complex roots. Page 344
Problem number : 6
Date solved : Monday, January 27, 2025 at 06:12:38 AM
CAS classification : system_of_ODEs

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

With initial conditions

\begin{align*} x_{1} \left (0\right ) = 1\\ x_{2} \left (0\right ) = 5 \end{align*}

✓ Solution by Maple

Time used: 0.014 (sec). Leaf size: 37

dsolve([diff(x__1(t),t) = 3*x__1(t)-2*x__2(t), diff(x__2(t),t) = 4*x__1(t)-x__2(t), x__1(0) = 1, x__2(0) = 5], singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{t} \left (-4 \sin \left (2 t \right )+\cos \left (2 t \right )\right ) \\ x_{2} \left (t \right ) &= {\mathrm e}^{t} \left (-3 \sin \left (2 t \right )+5 \cos \left (2 t \right )\right ) \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.007 (sec). Leaf size: 40

DSolve[{D[ x1[t],t]==3*x1[t]-2*x2[t],D[ x2[t],t]==4*x1[t]-1*x2[t]},{x1[0]==1,x2[0]==5},{x1[t],x2[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to e^t (\cos (2 t)-4 \sin (2 t)) \\ \text {x2}(t)\to e^t (5 \cos (2 t)-3 \sin (2 t)) \\ \end{align*}