14.32.5 problem 5

Internal problem ID [2829]
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 : 5
Date solved : Monday, January 27, 2025 at 06:19:05 AM
CAS classification : system_of_ODEs

\begin{align*} x_{1}^{\prime }\left (t \right )&=x_{1} \left (t \right )-4 x_{2} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=-8 x_{1} \left (t \right )+4 x_{2} \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.026 (sec). Leaf size: 85

dsolve([diff(x__1(t),t)=1*x__1(t)-4*x__2(t),diff(x__2(t),t)=-8*x__1(t)+4*x__2(t)],singsol=all)
 

\begin{align*} x_{1} \left (t \right ) &= c_1 \,{\mathrm e}^{\frac {\left (5+\sqrt {137}\right ) t}{2}}+c_2 \,{\mathrm e}^{-\frac {\left (-5+\sqrt {137}\right ) t}{2}} \\ x_{2} \left (t \right ) &= -\frac {c_1 \,{\mathrm e}^{\frac {\left (5+\sqrt {137}\right ) t}{2}} \sqrt {137}}{8}+\frac {c_2 \,{\mathrm e}^{-\frac {\left (-5+\sqrt {137}\right ) t}{2}} \sqrt {137}}{8}-\frac {3 c_1 \,{\mathrm e}^{\frac {\left (5+\sqrt {137}\right ) t}{2}}}{8}-\frac {3 c_2 \,{\mathrm e}^{-\frac {\left (-5+\sqrt {137}\right ) t}{2}}}{8} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.009 (sec). Leaf size: 148

DSolve[{D[x1[t],t]==1*x1[t]-4*x2[t],D[x2[t],t]==-8*x1[t]+4*x2[t]},{x1[t],x2[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x1}(t)\to \frac {1}{274} e^{-\frac {1}{2} \left (\sqrt {137}-5\right ) t} \left (c_1 \left (\left (137-3 \sqrt {137}\right ) e^{\sqrt {137} t}+137+3 \sqrt {137}\right )-8 \sqrt {137} c_2 \left (e^{\sqrt {137} t}-1\right )\right ) \\ \text {x2}(t)\to \frac {1}{274} e^{-\frac {1}{2} \left (\sqrt {137}-5\right ) t} \left (c_2 \left (\left (137+3 \sqrt {137}\right ) e^{\sqrt {137} t}+137-3 \sqrt {137}\right )-16 \sqrt {137} c_1 \left (e^{\sqrt {137} t}-1\right )\right ) \\ \end{align*}