76.6.16 problem 18

Internal problem ID [17471]
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 : 18
Date solved : Tuesday, January 28, 2025 at 10:39:12 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.075 (sec). Leaf size: 36

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

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

✓ Solution by Mathematica

Time used: 0.036 (sec). Leaf size: 81

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

\begin{align*} x(t)\to \frac {1}{4} e^{-t} \left (-12 e^t+(c_2-c_1) e^{4 t}+5 c_1-c_2\right ) \\ y(t)\to \frac {1}{4} e^{-t} \left (20 e^t-5 (c_1-c_2) e^{4 t}+5 c_1-c_2\right ) \\ \end{align*}