14.20.6 problem 6

Internal problem ID [2703]
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 : 6
Date solved : Monday, January 27, 2025 at 06:12:02 AM
CAS classification : system_of_ODEs

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

With initial conditions

\begin{align*} x \left (0\right ) = 0\\ y \left (0\right ) = 5 \end{align*}

✓ Solution by Maple

Time used: 0.026 (sec). Leaf size: 33

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

\begin{align*} x \left (t \right ) &= -3 \,{\mathrm e}^{4 t}+3 \,{\mathrm e}^{-t} \\ y &= 3 \,{\mathrm e}^{4 t}+2 \,{\mathrm e}^{-t} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.005 (sec). Leaf size: 37

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

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