71.19.3 problem 3

Internal problem ID [14595]
Book : Ordinary Differential Equations by Charles E. Roberts, Jr. CRC Press. 2010
Section : Chapter 10. Applications of Systems of Equations. Exercises 10.2 page 432
Problem number : 3
Date solved : Tuesday, January 28, 2025 at 06:44:03 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.046 (sec). Leaf size: 75

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

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

✓ Solution by Mathematica

Time used: 0.019 (sec). Leaf size: 96

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

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