76.6.12 problem 12

Internal problem ID [17467]
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 : 12
Date solved : Tuesday, January 28, 2025 at 10:39:08 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=x \left (t \right )-4 y \left (t \right )+2 t\\ \frac {d}{d t}y \left (t \right )&=x \left (t \right )-3 y \left (t \right )-3 \end{align*}

With initial conditions

\begin{align*} x \left (0\right ) = 1\\ y \left (0\right ) = -2 \end{align*}

✓ Solution by Maple

Time used: 0.083 (sec). Leaf size: 42

dsolve([diff(x(t),t) = x(t)-4*y(t)+2*t, diff(y(t),t) = x(t)-3*y(t)-3, x(0) = 1, y(0) = -2], singsol=all)
 

\begin{align*} x \left (t \right ) &= -{\mathrm e}^{-t}+2 \,{\mathrm e}^{-t} t +6 t +2 \\ y \left (t \right ) &= -{\mathrm e}^{-t}+{\mathrm e}^{-t} t -1+2 t \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.070 (sec). Leaf size: 46

DSolve[{D[x[t],t]==x[t]-4*y[t]+2*t,D[y[t],t]==x[t]-3*y[t]-3},{x[0]==1,y[0]==-2},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to e^{-t} \left (2 t+e^t (6 t+2)-1\right ) \\ y(t)\to e^{-t} \left (t+e^t (2 t-1)-1\right ) \\ \end{align*}