66.9.11 problem 25

Internal problem ID [16094]
Book : DIFFERENTIAL EQUATIONS by Paul Blanchard, Robert L. Devaney, Glen R. Hall. 4th edition. Brooks/Cole. Boston, USA. 2012
Section : Chapter 3. Linear Systems. Exercises section 3.1. page 258
Problem number : 25
Date solved : Thursday, October 02, 2025 at 10:42:25 AM
CAS classification : system_of_ODEs

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

With initial conditions

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

✓ Maple. Time used: 0.126 (sec). Leaf size: 25

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

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

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 26

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

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

✓ Sympy. Time used: 0.059 (sec). Leaf size: 37

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

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