57.22.6 problem 4(f)

Internal problem ID [14517]
Book : A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY. 2015.
Section : Chapter 4, Linear Systems. Exercises page 237
Problem number : 4(f)
Date solved : Thursday, October 02, 2025 at 09:38:04 AM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }&=3 x-2 y \left (t \right )\\ y^{\prime }\left (t \right )&=4 x-y \left (t \right ) \end{align*}

✓ Maple. Time used: 0.128 (sec). Leaf size: 52

ode:=[diff(x(t),t) = 3*x(t)-2*y(t), diff(y(t),t) = 4*x(t)-y(t)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 58

ode={D[x[t],t]==3*x[t]-2*y[t],D[y[t],t]==4*x[t]-y[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to e^t (c_1 \cos (2 t)+(c_1-c_2) \sin (2 t))\\ y(t)&\to e^t (c_2 \cos (2 t)+(2 c_1-c_2) \sin (2 t)) \end{align*}

✓ Sympy. Time used: 0.063 (sec). Leaf size: 54

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-3*x(t) + 2*y(t) + Derivative(x(t), t),0),Eq(-4*x(t) + y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

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