63.20.4 problem 2(d)

Internal problem ID [13226]
Book : A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY. 2015.
Section : Chapter 4, Linear Systems. Exercises page 218
Problem number : 2(d)
Date solved : Tuesday, January 28, 2025 at 05:12:42 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.037 (sec). Leaf size: 77

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

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

✓ Solution by Mathematica

Time used: 0.018 (sec). Leaf size: 94

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

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