70.1.19 problem 2.5

Internal problem ID [14309]
Book : Nonlinear Ordinary Differential Equations by D.W.Jordna and P.Smith. 4th edition 1999. Oxford Univ. Press. NY
Section : Chapter 2. Plane autonomous systems and linearization. Problems page 79
Problem number : 2.5
Date solved : Tuesday, January 28, 2025 at 06:26:14 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 49

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

\begin{align*} x \left (t \right ) &= c_{1} \sin \left (2 t \right )+c_{2} \cos \left (2 t \right ) \\ y \left (t \right ) &= -\frac {2 \cos \left (2 t \right ) c_{1}}{5}+\frac {2 \sin \left (2 t \right ) c_{2}}{5}+\frac {c_{1} \sin \left (2 t \right )}{5}+\frac {c_{2} \cos \left (2 t \right )}{5} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.005 (sec). Leaf size: 48

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

\begin{align*} x(t)\to c_1 \cos (2 t)+(c_1-5 c_2) \sin (t) \cos (t) \\ y(t)\to c_2 \cos (2 t)+(c_1-c_2) \sin (t) \cos (t) \\ \end{align*}