72.15.11 problem 19 (viii)

Internal problem ID [14892]
Book : DIFFERENTIAL EQUATIONS by Paul Blanchard, Robert L. Devaney, Glen R. Hall. 4th edition. Brooks/Cole. Boston, USA. 2012
Section : Chapter 3. Linear Systems. Review Exercises for chapter 3. page 376
Problem number : 19 (viii)
Date solved : Tuesday, January 28, 2025 at 07:18:13 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.055 (sec). Leaf size: 77

dsolve([diff(x(t),t)=-3*x(t)-3*y(t),diff(y(t),t)=2*x(t)+1*y(t)],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.015 (sec). Leaf size: 91

DSolve[{D[x[t],t]==-3*x[t]-3*y[t],D[y[t],t]==2*x[t]+1*y[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

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