Internal problem ID [5783]
Book: Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz.
McGraw-Hill NY. 2007. 1st Edition.
Section: Chapter 10. Systems of First-Order Equations. Section A. Drill exercises. Page
400
Problem number: 3(d).
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x^{\prime }\relax (t )&=x \relax (t )+2 y \relax (t )\\ y^{\prime }\relax (t )&=-4 x \relax (t )+y \relax (t ) \end {align*}
✓ Solution by Maple
Time used: 0.109 (sec). Leaf size: 57
dsolve([diff(x(t),t)=x(t)+2*y(t),diff(y(t),t)=-4*x(t)+y(t)],[x(t), y(t)], singsol=all)
\[ x \relax (t ) = -\frac {\sqrt {2}\, {\mathrm e}^{t} \left (\cos \left (2 t \sqrt {2}\right ) c_{1}-\sin \left (2 t \sqrt {2}\right ) c_{2}\right )}{2} \] \[ y \relax (t ) = {\mathrm e}^{t} \left (c_{2} \cos \left (2 t \sqrt {2}\right )+\sin \left (2 t \sqrt {2}\right ) c_{1}\right ) \]
✓ Solution by Mathematica
Time used: 0.02 (sec). Leaf size: 79
DSolve[{x'[t]==x[t]+2*y[t],y'[t]==-4*x[t]+y[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to c_1 e^t \cos \left (2 \sqrt {2} t\right )+\frac {c_2 e^t \sin \left (2 \sqrt {2} t\right )}{\sqrt {2}} \\ y(t)\to e^t \left (c_2 \cos \left (2 \sqrt {2} t\right )-\sqrt {2} c_1 \sin \left (2 \sqrt {2} t\right )\right ) \\ \end{align*}