79.12.1 problem (a)

Internal problem ID [21110]
Book : Ordinary Differential Equations. By Wolfgang Walter. Graduate texts in Mathematics. Springer. NY. QA372.W224 1998
Section : Chapter IV. Linear Differential Equations. Excercise XIII at page 189
Problem number : (a)
Date solved : Thursday, October 02, 2025 at 07:08:31 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=3 x \left (t \right )+6 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=-2 x \left (t \right )-3 y \left (t \right ) \end{align*}

✓ Maple. Time used: 0.055 (sec). Leaf size: 67

ode:=[diff(x(t),t) = 3*x(t)+6*y(t), diff(y(t),t) = -2*x(t)-3*y(t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= c_1 \sin \left (\sqrt {3}\, t \right )+c_2 \cos \left (\sqrt {3}\, t \right ) \\ y \left (t \right ) &= \frac {c_1 \sqrt {3}\, \cos \left (\sqrt {3}\, t \right )}{6}-\frac {c_2 \sqrt {3}\, \sin \left (\sqrt {3}\, t \right )}{6}-\frac {c_1 \sin \left (\sqrt {3}\, t \right )}{2}-\frac {c_2 \cos \left (\sqrt {3}\, t \right )}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.008 (sec). Leaf size: 77

ode={D[x[t],t]==3*x[t]+6*y[t], D[y[t],t]==-2*x[t]-3*y[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to c_1 \cos \left (\sqrt {3} t\right )+\sqrt {3} (c_1+2 c_2) \sin \left (\sqrt {3} t\right )\\ y(t)&\to c_2 \cos \left (\sqrt {3} t\right )-\frac {(2 c_1+3 c_2) \sin \left (\sqrt {3} t\right )}{\sqrt {3}} \end{align*}

✓ Sympy. Time used: 0.108 (sec). Leaf size: 61

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-3*x(t) - 6*y(t) + Derivative(x(t), t),0),Eq(-2*x(t) + 3*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = \frac {C_{1} \left (3 - \sqrt {21}\right ) e^{- \sqrt {21} t}}{2} + \frac {C_{2} \left (3 + \sqrt {21}\right ) e^{\sqrt {21} t}}{2}, \ y{\left (t \right )} = C_{1} e^{- \sqrt {21} t} + C_{2} e^{\sqrt {21} t}\right ] \]