4.19.8 problem 8

Internal problem ID [1449]
Book : Elementary differential equations and boundary value problems, 10th ed., Boyce and DiPrima
Section : Chapter 9.1, The Phase Plane: Linear Systems. page 505
Problem number : 8
Date solved : Tuesday, September 30, 2025 at 04:34:15 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=-x_{1} \left (t \right )-x_{2} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=-\frac {5 x_{2} \left (t \right )}{2} \end{align*}

✓ Maple. Time used: 0.147 (sec). Leaf size: 27

ode:=[diff(x__1(t),t) = -x__1(t)-x__2(t), diff(x__2(t),t) = -5/2*x__2(t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= \frac {2 c_2 \,{\mathrm e}^{-\frac {5 t}{2}}}{3}+c_1 \,{\mathrm e}^{-t} \\ x_{2} \left (t \right ) &= c_2 \,{\mathrm e}^{-\frac {5 t}{2}} \\ \end{align*}

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 47

ode={D[ x1[t],t]==-1*x1[t]-1*x2[t],D[ x2[t],t]==0*x1[t]-25/10*x2[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)&\to \left (c_1-\frac {2 c_2}{3}\right ) e^{-t}+\frac {2}{3} c_2 e^{-5 t/2}\\ \text {x2}(t)&\to c_2 e^{-5 t/2} \end{align*}

✓ Sympy. Time used: 0.046 (sec). Leaf size: 29

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
ode=[Eq(x__1(t) + x__2(t) + Derivative(x__1(t), t),0),Eq(5*x__2(t)/2 + Derivative(x__2(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
 

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