71.19.7 problem 7

Internal problem ID [14599]
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 : 7
Date solved : Tuesday, January 28, 2025 at 06:44:06 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=-5 x \left (t \right )-y \left (t \right )+2\\ \frac {d}{d t}y \left (t \right )&=3 x \left (t \right )-y \left (t \right )-3 \end{align*}

✓ Solution by Maple

Time used: 0.048 (sec). Leaf size: 38

dsolve([diff(x(t),t)=-5*x(t)-y(t)+2,diff(y(t),t)=3*x(t)-1*y(t)-3],singsol=all)
 

\begin{align*} x \left (t \right ) &= \frac {5}{8}-\frac {{\mathrm e}^{-4 t} c_{1}}{2}+{\mathrm e}^{-2 t} c_{2} \\ y \left (t \right ) &= \frac {{\mathrm e}^{-4 t} c_{1}}{2}-3 \,{\mathrm e}^{-2 t} c_{2} -\frac {9}{8} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.036 (sec). Leaf size: 93

DSolve[{D[x[t],t]==-5*x[t]-y[t]+2,D[y[t],t]==3*x[t]-1*y[t]-3},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to \frac {1}{48} e^{-4 t} \left (30 e^{4 t}-(1+24 c_1+24 c_2) e^{2 t}+3+72 c_1+24 c_2\right ) \\ y(t)\to \frac {1}{16} e^{-4 t} \left (-18 e^{4 t}+(1+24 c_1+24 c_2) e^{2 t}-1-24 c_1-8 c_2\right ) \\ \end{align*}