14.28.5 problem 10

Internal problem ID [2797]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 4. Qualitative theory of differential equations. Section 4.1 (Introduction). Page 377
Problem number : 10
Date solved : Monday, January 27, 2025 at 06:13:56 AM
CAS classification : system_of_ODEs

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

With initial conditions

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

✓ Solution by Maple

Time used: 0.023 (sec). Leaf size: 14

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

\begin{align*} x \left (t \right ) &= \cos \left (t \right ) \\ y &= \cos \left (t \right )+\sin \left (t \right ) \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.007 (sec). Leaf size: 15

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

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