Internal problem ID [12240]
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).
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x^{\prime }\left (t \right )&=2 x \left (t \right )+y \left (t \right )\\ y^{\prime }\left (t \right )&=y \left (t \right )-x \left (t \right ) \end {align*}
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 84
dsolve([diff(x(t),t)=2*x(t)+y(t),diff(y(t),t)=-x(t)+y(t)],[x(t), y(t)], singsol=all)
\[ x \left (t \right ) = \frac {{\mathrm e}^{\frac {3 t}{2}} \left (\sin \left (\frac {\sqrt {3}\, t}{2}\right ) \sqrt {3}\, c_{2} -\cos \left (\frac {\sqrt {3}\, t}{2}\right ) \sqrt {3}\, c_{1} -\sin \left (\frac {\sqrt {3}\, t}{2}\right ) c_{1} -\cos \left (\frac {\sqrt {3}\, t}{2}\right ) c_{2} \right )}{2} \] \[ y \left (t \right ) = {\mathrm e}^{\frac {3 t}{2}} \left (\sin \left (\frac {\sqrt {3}\, t}{2}\right ) c_{1} +\cos \left (\frac {\sqrt {3}\, t}{2}\right ) c_{2} \right ) \]
✓ Solution by Mathematica
Time used: 0.029 (sec). Leaf size: 111
DSolve[{x'[t]==2*x[t]+y[t],y'[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*}