50.29.3 problem 2(c)

Internal problem ID [8199]
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 : 2(c)
Date solved : Monday, January 27, 2025 at 03:45:43 PM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }\left (t \right )&=-3 x \left (t \right )+\sqrt {2}\, y \left (t \right )\\ y^{\prime }\left (t \right )&=\sqrt {2}\, x \left (t \right )-2 y \left (t \right ) \end{align*}

✓ Solution by Maple

Time used: 0.121 (sec). Leaf size: 39

dsolve([diff(x(t),t)=-3*x(t)+sqrt(2)*y(t),diff(y(t),t)=sqrt(2)*x(t)-2*y(t)],singsol=all)
 

\begin{align*} x \left (t \right ) &= {\mathrm e}^{-t} c_{1} +c_{2} {\mathrm e}^{-4 t} \\ y &= -\frac {\sqrt {2}\, \left (c_{2} {\mathrm e}^{-4 t}-2 \,{\mathrm e}^{-t} c_{1} \right )}{2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.004 (sec). Leaf size: 80

DSolve[{D[x[t],t]==-3*x[t]+Sqrt[2]*y[t],D[y[t],t]==Sqrt[2]*x[t]-2*y[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to \frac {1}{3} e^{-4 t} \left (c_1 \left (e^{3 t}+2\right )+\sqrt {2} c_2 \left (e^{3 t}-1\right )\right ) \\ y(t)\to \frac {1}{3} e^{-4 t} \left (\sqrt {2} c_1 \left (e^{3 t}-1\right )+c_2 \left (2 e^{3 t}+1\right )\right ) \\ \end{align*}