14.20.7 problem 7

Internal problem ID [2704]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 2. Second order differential equations. Section 2.14, The method of elimination for systems. Excercises page 258
Problem number : 7
Date solved : Monday, January 27, 2025 at 06:12:03 AM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )&=x \left (t \right )-y\\ y^{\prime }&=5 x \left (t \right )-3 y \end{align*}

With initial conditions

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

✓ Solution by Maple

Time used: 0.050 (sec). Leaf size: 29

dsolve([diff(x(t),t) = x(t)-y(t), diff(y(t),t) = 5*x(t)-3*y(t), x(0) = 1, y(0) = 2], singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.007 (sec). Leaf size: 29

DSolve[{D[x[t],t]==x[t]-y[t],D[y[t],t]==5*x[t]-3*y[t]},{x[0]==1,y[0]==2},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

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