63.19.1 problem 1(a)

Internal problem ID [13215]
Book : A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY. 2015.
Section : Chapter 4, Linear Systems. Exercises page 202
Problem number : 1(a)
Date solved : Tuesday, January 28, 2025 at 05:12:32 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.038 (sec). Leaf size: 94

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

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

✓ Solution by Mathematica

Time used: 0.014 (sec). Leaf size: 144

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

\begin{align*} x(t)\to -\frac {1}{4} e^{t-2 \sqrt {3} t} \left (c_1 \left (\left (\sqrt {3}-2\right ) e^{4 \sqrt {3} t}-2-\sqrt {3}\right )+\sqrt {3} c_2 \left (e^{4 \sqrt {3} t}-1\right )\right ) \\ y(t)\to \frac {1}{12} e^{t-2 \sqrt {3} t} \left (3 c_2 \left (\left (2+\sqrt {3}\right ) e^{4 \sqrt {3} t}+2-\sqrt {3}\right )-\sqrt {3} c_1 \left (e^{4 \sqrt {3} t}-1\right )\right ) \\ \end{align*}