70.1.3 problem 2.1 (iii)

Internal problem ID [14293]
Book : Nonlinear Ordinary Differential Equations by D.W.Jordna and P.Smith. 4th edition 1999. Oxford Univ. Press. NY
Section : Chapter 2. Plane autonomous systems and linearization. Problems page 79
Problem number : 2.1 (iii)
Date solved : Tuesday, January 28, 2025 at 06:25:53 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=-4 x \left (t \right )+2 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=3 x \left (t \right )-2 y \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.036 (sec). Leaf size: 82

dsolve([diff(x(t),t)=-4*x(t)+2*y(t),diff(y(t),t)=3*x(t)-2*y(t)],singsol=all)
 

\begin{align*} x \left (t \right ) &= c_{1} {\mathrm e}^{\left (-3+\sqrt {7}\right ) t}+c_{2} {\mathrm e}^{-\left (3+\sqrt {7}\right ) t} \\ y \left (t \right ) &= \frac {c_{1} {\mathrm e}^{\left (-3+\sqrt {7}\right ) t} \sqrt {7}}{2}-\frac {c_{2} {\mathrm e}^{-\left (3+\sqrt {7}\right ) t} \sqrt {7}}{2}+\frac {c_{1} {\mathrm e}^{\left (-3+\sqrt {7}\right ) t}}{2}+\frac {c_{2} {\mathrm e}^{-\left (3+\sqrt {7}\right ) t}}{2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.008 (sec). Leaf size: 143

DSolve[{D[x[t],t]==-4*x[t]+2*y[t],D[y[t],t]==3*x[t]-2*y[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to \frac {1}{14} e^{-\left (\left (3+\sqrt {7}\right ) t\right )} \left (c_1 \left (-\left (\sqrt {7}-7\right ) e^{2 \sqrt {7} t}+7+\sqrt {7}\right )+2 \sqrt {7} c_2 \left (e^{2 \sqrt {7} t}-1\right )\right ) \\ y(t)\to \frac {1}{14} e^{-\left (\left (3+\sqrt {7}\right ) t\right )} \left (3 \sqrt {7} c_1 \left (e^{2 \sqrt {7} t}-1\right )+c_2 \left (\left (7+\sqrt {7}\right ) e^{2 \sqrt {7} t}+7-\sqrt {7}\right )\right ) \\ \end{align*}