Internal problem ID [12245]
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: 2.2 (v).
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x^{\prime }\left (t \right )&=-y \left (t \right )\\ y^{\prime }\left (t \right )&=2 x \left (t \right )-4 y \left (t \right ) \end {align*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 81
dsolve([diff(x(t),t)=-y(t),diff(y(t),t)=2*x(t)-4*y(t)],[x(t), y(t)], singsol=all)
\[ x \left (t \right ) = -\frac {c_{2} {\mathrm e}^{-\left (2+\sqrt {2}\right ) t} \sqrt {2}}{2}+\frac {c_{1} {\mathrm e}^{\left (-2+\sqrt {2}\right ) t} \sqrt {2}}{2}+c_{2} {\mathrm e}^{-\left (2+\sqrt {2}\right ) t}+c_{1} {\mathrm e}^{\left (-2+\sqrt {2}\right ) t} \] \[ y \left (t \right ) = c_{1} {\mathrm e}^{\left (-2+\sqrt {2}\right ) t}+c_{2} {\mathrm e}^{-\left (2+\sqrt {2}\right ) t} \]
✓ Solution by Mathematica
Time used: 0.019 (sec). Leaf size: 143
DSolve[{x'[t]==-y[t],y'[t]==2*x[t]-4*y[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to \frac {1}{4} e^{-\left (\left (2+\sqrt {2}\right ) t\right )} \left (2 c_1 \left (\left (1+\sqrt {2}\right ) e^{2 \sqrt {2} t}+1-\sqrt {2}\right )-\sqrt {2} c_2 \left (e^{2 \sqrt {2} t}-1\right )\right ) y(t)\to \frac {1}{2} e^{-\left (\left (2+\sqrt {2}\right ) t\right )} \left (\sqrt {2} c_1 \left (e^{2 \sqrt {2} t}-1\right )+c_2 \left (-\left (\sqrt {2}-1\right ) e^{2 \sqrt {2} t}+1+\sqrt {2}\right )\right ) \end{align*}