70.7.14 problem 14

Internal problem ID [18774]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 3. Systems of two first order equations. Section 3.3 (Homogeneous linear systems with constant coefficients). Problems at page 165
Problem number : 14
Date solved : Thursday, October 02, 2025 at 03:30:45 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=2 x \left (t \right )-y \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=3 x \left (t \right )-2 y \left (t \right ) \end{align*}

With initial conditions

\begin{align*} x \left (0\right )&=2 \\ y \left (0\right )&=5 \\ \end{align*}

✓ Maple. Time used: 0.116 (sec). Leaf size: 29

ode:=[diff(x(t),t) = 2*x(t)-y(t), diff(y(t),t) = 3*x(t)-2*y(t)]; 
ic:=[x(0) = 2, y(0) = 5]; 
dsolve([ode,op(ic)]);
 

\begin{align*} x \left (t \right ) &= \frac {{\mathrm e}^{t}}{2}+\frac {3 \,{\mathrm e}^{-t}}{2} \\ y \left (t \right ) &= \frac {{\mathrm e}^{t}}{2}+\frac {9 \,{\mathrm e}^{-t}}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 40

ode={D[x[t],t]==2*x[t]-y[t],D[y[t],t]==3*x[t]-2*y[t]}; 
ic={x[0]==2,y[0]==5}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.043 (sec). Leaf size: 26

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

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