10.17.10 problem 10

Internal problem ID [1425]
Book : Elementary differential equations and boundary value problems, 10th ed., Boyce and DiPrima
Section : Chapter 7.8, Repeated Eigenvalues. page 436
Problem number : 10
Date solved : Tuesday, March 04, 2025 at 12:35:40 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=3 x_{1} \left (t \right )+9 x_{2} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=-x_{1} \left (t \right )-3 x_{2} \left (t \right ) \end{align*}

With initial conditions

\begin{align*} x_{1} \left (0\right ) = 2\\ x_{2} \left (0\right ) = 4 \end{align*}

✓ Maple. Time used: 0.015 (sec). Leaf size: 17

ode:=[diff(x__1(t),t) = 3*x__1(t)+9*x__2(t), diff(x__2(t),t) = -x__1(t)-3*x__2(t)]; 
ic:=x__1(0) = 2x__2(0) = 4; 
dsolve([ode,ic]);
 

\begin{align*} x_{1} \left (t \right ) &= 42 t +2 \\ x_{2} \left (t \right ) &= 4-14 t \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 18

ode={D[ x1[t],t]==3*x1[t]+9*x2[t],D[ x2[t],t]==-1*x1[t]-3*x2[t]}; 
ic={x1[0]==2,x2[0]==4}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to 42 t+2 \\ \text {x2}(t)\to 4-14 t \\ \end{align*}

✓ Sympy. Time used: 0.064 (sec). Leaf size: 20

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
ode=[Eq(-3*x__1(t) - 9*x__2(t) + Derivative(x__1(t), t),0),Eq(x__1(t) + 3*x__2(t) + Derivative(x__2(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
 

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