14.32.10 problem 10

Internal problem ID [2834]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 4. Qualitative theory of differential equations. Section 4.7 (Phase portraits of linear systems). Page 427
Problem number : 10
Date solved : Monday, January 27, 2025 at 06:19:09 AM
CAS classification : system_of_ODEs

\begin{align*} x_{1}^{\prime }\left (t \right )&=4 x_{2} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=-9 x_{1} \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.016 (sec). Leaf size: 35

dsolve([diff(x__1(t),t)=0*x__1(t)+4*x__2(t),diff(x__2(t),t)=-9*x__1(t)-0*x__2(t)],singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= c_1 \sin \left (6 t \right )+c_2 \cos \left (6 t \right ) \\ x_{2} \left (t \right ) &= \frac {3 c_1 \cos \left (6 t \right )}{2}-\frac {3 c_2 \sin \left (6 t \right )}{2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.005 (sec). Leaf size: 44

DSolve[{D[x1[t],t]==0*x1[t]+4*x2[t],D[x2[t],t]==-9*x1[t]-0*x2[t]},{x1[t],x2[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to c_1 \cos (6 t)+\frac {2}{3} c_2 \sin (6 t) \\ \text {x2}(t)\to c_2 \cos (6 t)-\frac {3}{2} c_1 \sin (6 t) \\ \end{align*}