Internal problem ID [5774]
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 10.3 Homogeneous Linear Systems
with Constant Coefficients. Page 387
Problem number: 1(h).
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 )+5 y \relax (t ) \end {align*}
✓ Solution by Maple
Time used: 0.062 (sec). Leaf size: 58
dsolve([diff(x(t),t)=x(t)-2*y(t),diff(y(t),t)=4*x(t)+5*y(t)],[x(t), y(t)], singsol=all)
\[ x \relax (t ) = -\frac {{\mathrm e}^{3 t} \left (\sin \left (2 t \right ) c_{1}+\sin \left (2 t \right ) c_{2}-\cos \left (2 t \right ) c_{1}+\cos \left (2 t \right ) c_{2}\right )}{2} \] \[ y \relax (t ) = {\mathrm e}^{3 t} \left (\sin \left (2 t \right ) c_{1}+\cos \left (2 t \right ) c_{2}\right ) \]
✓ Solution by Mathematica
Time used: 0.005 (sec). Leaf size: 59
DSolve[{x'[t]==x[t]-2*y[t],y'[t]==4*x[t]+5*y[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to e^{3 t} (c_1 \cos (2 t)-(c_1+c_2) \sin (2 t)) \\ y(t)\to e^{3 t} (c_2 \cos (2 t)+(2 c_1+c_2) \sin (2 t)) \\ \end{align*}