Internal problem ID [11559]
Book: A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag,
NY. 2015.
Section: Chapter 4, Linear Systems. Exercises page 218
Problem number: 2(c).
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x^{\prime }&=-x+y \left (t \right )\\ y^{\prime }\left (t \right )&=x-2 y \left (t \right ) \end {align*}
✓ Solution by Maple
Time used: 0.079 (sec). Leaf size: 86
dsolve([diff(x(t),t)=-x(t)+y(t),diff(y(t),t)=x(t)-2*y(t)],[x(t), y(t)], singsol=all)
\[ x \left (t \right ) = \frac {c_{1} {\mathrm e}^{\frac {\left (\sqrt {5}-3\right ) t}{2}} \sqrt {5}}{2}-\frac {c_{2} {\mathrm e}^{-\frac {\left (\sqrt {5}+3\right ) t}{2}} \sqrt {5}}{2}+\frac {c_{1} {\mathrm e}^{\frac {\left (\sqrt {5}-3\right ) t}{2}}}{2}+\frac {c_{2} {\mathrm e}^{-\frac {\left (\sqrt {5}+3\right ) t}{2}}}{2} \] \[ y \left (t \right ) = c_{1} {\mathrm e}^{\frac {\left (\sqrt {5}-3\right ) t}{2}}+c_{2} {\mathrm e}^{-\frac {\left (\sqrt {5}+3\right ) t}{2}} \]
✓ Solution by Mathematica
Time used: 0.015 (sec). Leaf size: 145
DSolve[{x'[t]==-x[t]+y[t],y'[t]==x[t]-2*y[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to \frac {1}{10} e^{-\frac {1}{2} \left (3+\sqrt {5}\right ) t} \left (c_1 \left (\left (5+\sqrt {5}\right ) e^{\sqrt {5} t}+5-\sqrt {5}\right )+2 \sqrt {5} c_2 \left (e^{\sqrt {5} t}-1\right )\right ) y(t)\to \frac {1}{10} e^{-\frac {1}{2} \left (3+\sqrt {5}\right ) t} \left (2 \sqrt {5} c_1 \left (e^{\sqrt {5} t}-1\right )-c_2 \left (\left (\sqrt {5}-5\right ) e^{\sqrt {5} t}-5-\sqrt {5}\right )\right ) \end{align*}