14.32.7 problem 7

Internal problem ID [2831]
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 : 7
Date solved : Monday, January 27, 2025 at 06:19:07 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.023 (sec). Leaf size: 81

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

\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{-\frac {t}{2}} \left (\sin \left (\frac {\sqrt {15}\, t}{2}\right ) c_1 +\cos \left (\frac {\sqrt {15}\, t}{2}\right ) c_2 \right ) \\ x_{2} \left (t \right ) &= -\frac {{\mathrm e}^{-\frac {t}{2}} \left (\sin \left (\frac {\sqrt {15}\, t}{2}\right ) \sqrt {15}\, c_2 -\cos \left (\frac {\sqrt {15}\, t}{2}\right ) \sqrt {15}\, c_1 +\sin \left (\frac {\sqrt {15}\, t}{2}\right ) c_1 +\cos \left (\frac {\sqrt {15}\, t}{2}\right ) c_2 \right )}{4} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.023 (sec). Leaf size: 111

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

\begin{align*} \text {x1}(t)\to \frac {1}{15} e^{-t/2} \left (15 c_1 \cos \left (\frac {\sqrt {15} t}{2}\right )+\sqrt {15} (c_1+4 c_2) \sin \left (\frac {\sqrt {15} t}{2}\right )\right ) \\ \text {x2}(t)\to \frac {1}{15} e^{-t/2} \left (15 c_2 \cos \left (\frac {\sqrt {15} t}{2}\right )-\sqrt {15} (4 c_1+c_2) \sin \left (\frac {\sqrt {15} t}{2}\right )\right ) \\ \end{align*}