87.19.2 problem 2 (c)

Internal problem ID [23672]
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 149
Problem number : 2 (c)
Date solved : Thursday, October 02, 2025 at 09:44:08 PM
CAS classification : system_of_ODEs

\begin{align*} x_{1}^{\prime }\left (t \right )&=-3 x_{1} \left (t \right )+4 x_{2} \left (t \right )\\ x_{2}^{\prime }\left (t \right )&=-2 x_{1} \left (t \right )+3 x_{2} \left (t \right ) \end{align*}

With initial conditions

\begin{align*} x_{1} \left (0\right )&=-1 \\ x_{2} \left (0\right )&=3 \\ \end{align*}

✓ Maple. Time used: 0.049 (sec). Leaf size: 29

ode:=[diff(x__1(t),t) = -3*x__1(t)+4*x__2(t), diff(x__2(t),t) = -2*x__1(t)+3*x__2(t)]; 
ic:=[x__1(0) = -1, x__2(0) = 3]; 
dsolve([ode,op(ic)]);
 

\begin{align*} x_{1} \left (t \right ) &= -8 \,{\mathrm e}^{-t}+7 \,{\mathrm e}^{t} \\ x_{2} \left (t \right ) &= -4 \,{\mathrm e}^{-t}+7 \,{\mathrm e}^{t} \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 34

ode={D[x1[t],t]==-3*x1[t]+4*x2[t],D[x2[t],t]==-2*x1[t]+3*x2[t]}; 
ic={x1[0]==-1,x2[0]==3}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)&\to 7 e^t-8 e^{-t}\\ \text {x2}(t)&\to 7 e^t-4 e^{-t} \end{align*}

✓ Sympy. Time used: 0.065 (sec). Leaf size: 24

from sympy import * 
t = symbols("t") 
x1 = Function("x1") 
x2 = Function("x2") 
ode=[Eq(3*x1(t) - 4*x2(t) + Derivative(x1(t), t),0),Eq(2*x1(t) - 3*x2(t) + Derivative(x2(t), t),0)] 
ics = {x1(0): -1, x2(0): 3} 
dsolve(ode,func=[x1(t),x2(t)],ics=ics)
 

\[ \left [ x_{1}{\left (t \right )} = 7 e^{t} - 8 e^{- t}, \ x_{2}{\left (t \right )} = 7 e^{t} - 4 e^{- t}\right ] \]