73.27.11 problem 38.10 (e)

Internal problem ID [15767]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 38. Systems of differential equations. A starting point. Additional Exercises. page 786
Problem number : 38.10 (e)
Date solved : Tuesday, January 28, 2025 at 08:06:51 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=4 x \left (t \right )-13 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=x \left (t \right ) \end{align*}

With initial conditions

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

✓ Solution by Maple

Time used: 0.047 (sec). Leaf size: 34

dsolve([diff(x(t),t) = 4*x(t)-13*y(t), diff(y(t),t) = x(t), x(0) = 2, y(0) = 1], singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.007 (sec). Leaf size: 37

DSolve[{D[x[t],t]==4*x[t]-13*y[t],D[y[t],t]==x[t]},{x[0]==2,y[0]==1},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

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