76.9.7 problem 7

Internal problem ID [17517]
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 3.5 (Repeated Eigenvalues). Problems at page 188
Problem number : 7
Date solved : Tuesday, January 28, 2025 at 10:39:48 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 )\\ \frac {d}{d t}y \left (t \right )&=4 x \left (t \right )-7 y \left (t \right ) \end{align*}

With initial conditions

\begin{align*} x \left (0\right ) = 3\\ y \left (0\right ) = 2 \end{align*}

✓ Solution by Maple

Time used: 0.075 (sec). Leaf size: 28

dsolve([diff(x(t),t) = x(t)-4*y(t), diff(y(t),t) = 4*x(t)-7*y(t), x(0) = 3, y(0) = 2], singsol=all)
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{-3 t} \left (4 t +3\right ) \\ y \left (t \right ) &= \frac {{\mathrm e}^{-3 t} \left (16 t +8\right )}{4} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 30

DSolve[{D[x[t],t]==x[t]-4*y[t],D[y[t],t]==4*x[t]-7*y[t]},{x[0]==3,y[0]==2},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to e^{-3 t} (4 t+3) \\ y(t)\to e^{-3 t} (4 t+2) \\ \end{align*}