66.10.14 problem 12 (a)

Internal problem ID [16112]
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 3.2. page 277
Problem number : 12 (a)
Date solved : Thursday, October 02, 2025 at 10:42:35 AM
CAS classification : system_of_ODEs

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

With initial conditions

\begin{align*} x \left (0\right )&=1 \\ y \left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.159 (sec). Leaf size: 24

ode:=[diff(x(t),t) = 3*x(t), diff(y(t),t) = x(t)-2*y(t)]; 
ic:=[x(0) = 1, y(0) = 0]; 
dsolve([ode,op(ic)]);
 

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

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 29

ode={D[x[t],t]==3*x[t],D[y[t],t]==x[t]-2*y[t]}; 
ic={x[0]==1,y[0]==0}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.044 (sec). Leaf size: 26

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

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