63.23.2 problem 4

Internal problem ID [13242]
Book : A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY. 2015.
Section : Chapter 4, Linear Systems. Exercises page 244
Problem number : 4
Date solved : Tuesday, January 28, 2025 at 05:12:55 AM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }&=-5 x+3 y \left (t \right )+{\mathrm e}^{-t}\\ y^{\prime }\left (t \right )&=2 x-10 y \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.049 (sec). Leaf size: 47

dsolve([diff(x(t),t)=-5*x(t)+3*y(t)+exp(-t),diff(y(t),t)=2*x(t)-10*y(t)],singsol=all)
 

\begin{align*} x \left (t \right ) &= 3 c_{2} {\mathrm e}^{-4 t}-\frac {{\mathrm e}^{-11 t} c_{1}}{2}+\frac {3 \,{\mathrm e}^{-t}}{10} \\ y &= c_{2} {\mathrm e}^{-4 t}+{\mathrm e}^{-11 t} c_{1} +\frac {{\mathrm e}^{-t}}{15} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.059 (sec). Leaf size: 88

DSolve[{D[x[t],t]==-5*x[t]+3*y[t]+Exp[-t],D[y[t],t]==2*x[t]-10*y[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to \frac {1}{70} e^{-11 t} \left (21 e^{10 t}+30 (2 c_1+c_2) e^{7 t}+10 (c_1-3 c_2)\right ) \\ y(t)\to \frac {1}{105} e^{-11 t} \left (7 e^{10 t}+15 (2 c_1+c_2) e^{7 t}-30 (c_1-3 c_2)\right ) \\ \end{align*}