44.29.3 problem 2(c)

Internal problem ID [9485]
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 : Tuesday, September 30, 2025 at 06:19:21 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.155 (sec). Leaf size: 40

ode:=[diff(x(t),t) = -3*x(t)+2^(1/2)*y(t), diff(y(t),t) = 2^(1/2)*x(t)-2*y(t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= c_1 \,{\mathrm e}^{-t}+c_2 \,{\mathrm e}^{-4 t} \\ y \left (t \right ) &= \frac {\left (2 c_1 \,{\mathrm e}^{-t}-c_2 \,{\mathrm e}^{-4 t}\right ) \sqrt {2}}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 80

ode={D[x[t],t]==-3*x[t]+Sqrt[2]*y[t],D[y[t],t]==Sqrt[2]*x[t]-2*y[t]}; 
ic={}; 
DSolve[{ode,ic},{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*}

✓ Sympy. Time used: 0.096 (sec). Leaf size: 39

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(3*x(t) - sqrt(2)*y(t) + Derivative(x(t), t),0),Eq(-sqrt(2)*x(t) + 2*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = - \sqrt {2} C_{1} e^{- 4 t} + \frac {\sqrt {2} C_{2} e^{- t}}{2}, \ y{\left (t \right )} = C_{1} e^{- 4 t} + C_{2} e^{- t}\right ] \]