57.20.4 problem 2(d)

Internal problem ID [14507]
Book : A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY. 2015.
Section : Chapter 4, Linear Systems. Exercises page 218
Problem number : 2(d)
Date solved : Thursday, October 02, 2025 at 09:37:59 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.133 (sec). Leaf size: 77

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

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

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 94

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

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

✓ Sympy. Time used: 0.117 (sec). Leaf size: 85

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

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