3.4.3 problem problem 3

Internal problem ID [967]
Book : Differential equations and linear algebra, 4th ed., Edwards and Penney
Section : Section 7.3, The eigenvalue method for linear systems. Page 395
Problem number : problem 3
Date solved : Tuesday, September 30, 2025 at 04:20:03 AM
CAS classification : system_of_ODEs

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

With initial conditions

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

✓ Maple. Time used: 0.131 (sec). Leaf size: 33

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

\begin{align*} x_{1} \left (t \right ) &= \frac {8 \,{\mathrm e}^{6 t}}{7}-\frac {{\mathrm e}^{-t}}{7} \\ x_{2} \left (t \right ) &= \frac {6 \,{\mathrm e}^{6 t}}{7}+\frac {{\mathrm e}^{-t}}{7} \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 44

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

\begin{align*} \text {x1}(t)&\to \frac {1}{7} e^{-t} \left (8 e^{7 t}-1\right )\\ \text {x2}(t)&\to \frac {1}{7} e^{-t} \left (6 e^{7 t}+1\right ) \end{align*}

✓ Sympy. Time used: 0.052 (sec). Leaf size: 31

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
ode=[Eq(-3*x__1(t) - 4*x__2(t) + Derivative(x__1(t), t),0),Eq(-3*x__1(t) - 2*x__2(t) + Derivative(x__2(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
 

\[ \left [ x^{1}{\left (t \right )} = - C_{1} e^{- t} + \frac {4 C_{2} e^{6 t}}{3}, \ x^{2}{\left (t \right )} = C_{1} e^{- t} + C_{2} e^{6 t}\right ] \]