57.23.5 problem 7

Internal problem ID [14526]
Book : A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY. 2015.
Section : Chapter 4, Linear Systems. Exercises page 244
Problem number : 7
Date solved : Thursday, October 02, 2025 at 09:38:11 AM
CAS classification : system_of_ODEs

\begin{align*} x^{\prime }&=x-3 y \left (t \right )\\ y^{\prime }\left (t \right )&=3 x+7 y \left (t \right ) \end{align*}

✓ Maple. Time used: 0.115 (sec). Leaf size: 32

ode:=[diff(x(t),t) = x(t)-3*y(t), diff(y(t),t) = 3*x(t)+7*y(t)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 46

ode={D[x[t],t]==x[t]-3*y[t],D[y[t],t]==3*x[t]+7*y[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to -e^{4 t} (c_1 (3 t-1)+3 c_2 t)\\ y(t)&\to e^{4 t} (3 (c_1+c_2) t+c_2) \end{align*}

✓ Sympy. Time used: 0.062 (sec). Leaf size: 44

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-x(t) + 3*y(t) + Derivative(x(t), t),0),Eq(-3*x(t) - 7*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

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