65.19.3 problem 3

Internal problem ID [15882]
Book : Ordinary Differential Equations by Charles E. Roberts, Jr. CRC Press. 2010
Section : Chapter 10. Applications of Systems of Equations. Exercises 10.2 page 432
Problem number : 3
Date solved : Thursday, October 02, 2025 at 10:29:07 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=-x \left (t \right )-2 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: 0.139 (sec). Leaf size: 75

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

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

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 96

ode={D[x[t],t]==-x[t]-2*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 \frac {1}{3} e^{-2 t} \left (3 c_1 \cos \left (\sqrt {3} t\right )+\sqrt {3} (c_1-2 c_2) \sin \left (\sqrt {3} t\right )\right )\\ y(t)&\to \frac {1}{3} e^{-2 t} \left (3 c_2 \cos \left (\sqrt {3} t\right )+\sqrt {3} (2 c_1-c_2) \sin \left (\sqrt {3} t\right )\right ) \end{align*}

✓ Sympy. Time used: 0.113 (sec). Leaf size: 85

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(x(t) + 2*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)
 

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