57.20.2 problem 2(b)

Internal problem ID [14505]
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(b)
Date solved : Thursday, October 02, 2025 at 09:37:58 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.139 (sec). Leaf size: 23

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

\begin{align*} x \left (t \right ) &= c_2 \,{\mathrm e}^{t} \\ y \left (t \right ) &= \frac {3 c_2 \,{\mathrm e}^{t}}{5}+c_1 \,{\mathrm e}^{-4 t} \\ \end{align*}

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 149

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

\begin{align*} x(t)&\to \frac {1}{74} e^{-\frac {1}{2} \left (3+\sqrt {37}\right ) t} \left (c_1 \left (\left (37+5 \sqrt {37}\right ) e^{\sqrt {37} t}+37-5 \sqrt {37}\right )+2 \sqrt {37} c_2 \left (e^{\sqrt {37} t}-1\right )\right )\\ y(t)&\to \frac {1}{74} e^{-\frac {1}{2} \left (3+\sqrt {37}\right ) t} \left (6 \sqrt {37} c_1 \left (e^{\sqrt {37} t}-1\right )-c_2 \left (\left (5 \sqrt {37}-37\right ) e^{\sqrt {37} t}-37-5 \sqrt {37}\right )\right ) \end{align*}

✓ Sympy. Time used: 0.042 (sec). Leaf size: 24

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

\[ \left [ x{\left (t \right )} = \frac {5 C_{1} e^{t}}{3}, \ y{\left (t \right )} = C_{1} e^{t} + C_{2} e^{- 4 t}\right ] \]