67.27.2 problem 38.2

Internal problem ID [17045]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 38. Systems of differential equations. A starting point. Additional Exercises. page 786
Problem number : 38.2
Date solved : Thursday, October 02, 2025 at 01:42:18 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=4 x \left (t \right )-3 y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=6 x \left (t \right )-7 y \left (t \right ) \end{align*}

✓ Maple. Time used: 0.110 (sec). Leaf size: 35

ode:=[diff(x(t),t) = 4*x(t)-3*y(t), diff(y(t),t) = 6*x(t)-7*y(t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= c_1 \,{\mathrm e}^{2 t}+c_2 \,{\mathrm e}^{-5 t} \\ y \left (t \right ) &= \frac {2 c_1 \,{\mathrm e}^{2 t}}{3}+3 c_2 \,{\mathrm e}^{-5 t} \\ \end{align*}

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 74

ode={D[x[t],t]==4*x[t]-3*y[t],D[y[t],t]==6*x[t]-7*y[t]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {1}{7} e^{-5 t} \left (c_1 \left (9 e^{7 t}-2\right )-3 c_2 \left (e^{7 t}-1\right )\right )\\ y(t)&\to \frac {1}{7} e^{-5 t} \left (6 c_1 \left (e^{7 t}-1\right )+c_2 \left (9-2 e^{7 t}\right )\right ) \end{align*}

✓ Sympy. Time used: 0.057 (sec). Leaf size: 36

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-4*x(t) + 3*y(t) + Derivative(x(t), t),0),Eq(-6*x(t) + 7*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

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