71.19.1 problem 1

Internal problem ID [14593]
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 : 1
Date solved : Tuesday, January 28, 2025 at 06:44:01 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.027 (sec). Leaf size: 30

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

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

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 72

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

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