76.6.18 problem 20

Internal problem ID [17473]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 3. Systems of two first order equations. Section 3.2 (Two first order linear differential equations). Problems at page 142
Problem number : 20
Date solved : Tuesday, January 28, 2025 at 10:39:14 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=-5 x \left (t \right )+4 y \left (t \right )-35\\ \frac {d}{d t}y \left (t \right )&=-2 x \left (t \right )+y \left (t \right )-11 \end{align*}

✓ Solution by Maple

Time used: 0.074 (sec). Leaf size: 36

dsolve([diff(x(t),t)=-5*x(t)+4*y(t)-35,diff(y(t),t)=-2*x(t)+y(t)-11],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.013 (sec). Leaf size: 72

DSolve[{D[x[t],t]==-5*x[t]+4*y[t]-35,D[y[t],t]==-2*x[t]+y[t]-11},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to (c_1-2 c_2) \left (-e^{-t}\right )+2 (c_1-c_2) e^{-3 t}-3 \\ y(t)\to e^{-3 t} \left (5 e^{3 t}-(c_1-2 c_2) e^{2 t}+c_1-c_2\right ) \\ \end{align*}