70.1.6 problem 2.1 (vi)

Internal problem ID [14296]
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 : 2.1 (vi)
Date solved : Tuesday, January 28, 2025 at 06:25:55 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=2 x \left (t \right )+y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=y \left (t \right )-x \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.046 (sec). Leaf size: 81

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

\begin{align*} x \left (t \right ) &= {\mathrm e}^{\frac {3 t}{2}} \left (\sin \left (\frac {\sqrt {3}\, t}{2}\right ) c_{1} +\cos \left (\frac {\sqrt {3}\, t}{2}\right ) c_{2} \right ) \\ y \left (t \right ) &= -\frac {{\mathrm e}^{\frac {3 t}{2}} \left (\sqrt {3}\, \sin \left (\frac {\sqrt {3}\, t}{2}\right ) c_{2} -\sqrt {3}\, \cos \left (\frac {\sqrt {3}\, t}{2}\right ) c_{1} +\sin \left (\frac {\sqrt {3}\, t}{2}\right ) c_{1} +\cos \left (\frac {\sqrt {3}\, t}{2}\right ) c_{2} \right )}{2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.016 (sec). Leaf size: 111

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

\begin{align*} x(t)\to \frac {1}{3} e^{3 t/2} \left (3 c_1 \cos \left (\frac {\sqrt {3} t}{2}\right )+\sqrt {3} (c_1+2 c_2) \sin \left (\frac {\sqrt {3} t}{2}\right )\right ) \\ y(t)\to \frac {1}{3} e^{3 t/2} \left (3 c_2 \cos \left (\frac {\sqrt {3} t}{2}\right )-\sqrt {3} (2 c_1+c_2) \sin \left (\frac {\sqrt {3} t}{2}\right )\right ) \\ \end{align*}