87.20.11 problem 12

Internal problem ID [23696]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 3. Linear Systems. Exercise at page 161
Problem number : 12
Date solved : Thursday, October 02, 2025 at 09:44:21 PM
CAS classification : system_of_ODEs

\begin{align*} x_{1}^{\prime }\left (t \right )&=3 x_{1} \left (t \right )-2 x_{2} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=x_{1} \left (t \right ) \end{align*}

With initial conditions

\begin{align*} x_{1} \left (0\right )&=-1 \\ x_{2} \left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.048 (sec). Leaf size: 25

ode:=[diff(x__1(t),t) = 3*x__1(t)-2*x__2(t), diff(x__2(t),t) = x__1(t)]; 
ic:=[x__1(0) = -1, x__2(0) = 0]; 
dsolve([ode,op(ic)]);
 

\begin{align*} x_{1} \left (t \right ) &= -2 \,{\mathrm e}^{2 t}+{\mathrm e}^{t} \\ x_{2} \left (t \right ) &= -{\mathrm e}^{2 t}+{\mathrm e}^{t} \\ \end{align*}

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

ode={D[x1[t],t]==3*x1[t]-2*x2[t],D[x2[t],t]==x1[t]}; 
ic={x1[0]==-1,x2[0]==0}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)&\to e^t-2 e^{2 t}\\ \text {x2}(t)&\to -e^t \left (e^t-1\right ) \end{align*}

✓ Sympy. Time used: 0.061 (sec). Leaf size: 22

from sympy import * 
t = symbols("t") 
x1 = Function("x1") 
x2 = Function("x2") 
ode=[Eq(-3*x1(t) + 2*x2(t) + Derivative(x1(t), t),0),Eq(-x1(t) + Derivative(x2(t), t),0)] 
ics = {x1(0): -1, x2(0): 0} 
dsolve(ode,func=[x1(t),x2(t)],ics=ics)
 

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