10.17.11 problem 11

Internal problem ID [1426]
Book : Elementary differential equations and boundary value problems, 10th ed., Boyce and DiPrima
Section : Chapter 7.8, Repeated Eigenvalues. page 436
Problem number : 11
Date solved : Tuesday, March 04, 2025 at 12:35:41 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=x_{1} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=-4 x_{1} \left (t \right )+x_{2} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=3 x_{1} \left (t \right )+6 x_{2} \left (t \right )+2 x_{3} \left (t \right ) \end{align*}

With initial conditions

\begin{align*} x_{1} \left (0\right ) = -1\\ x_{2} \left (0\right ) = 2\\ x_{3} \left (0\right ) = -30 \end{align*}

✓ Maple. Time used: 0.044 (sec). Leaf size: 38

ode:=[diff(x__1(t),t) = x__1(t), diff(x__2(t),t) = -4*x__1(t)+x__2(t), diff(x__3(t),t) = 3*x__1(t)+6*x__2(t)+2*x__3(t)]; 
ic:=x__1(0) = -1x__2(0) = 2x__3(0) = -30; 
dsolve([ode,ic]);
 

\begin{align*} x_{1} \left (t \right ) &= -{\mathrm e}^{t} \\ x_{2} \left (t \right ) &= \left (4 t +2\right ) {\mathrm e}^{t} \\ x_{3} \left (t \right ) &= -24 t \,{\mathrm e}^{t}-33 \,{\mathrm e}^{t}+3 \,{\mathrm e}^{2 t} \\ \end{align*}

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 39

ode={D[ x1[t],t]==1*x1[t]+0*x2[t]+0*x3[t],D[ x2[t],t]==-4*x1[t]+1*x2[t]+0*x3[t],D[ x3[t],t]==3*x1[t]+6*x2[t]+2*x3[t]}; 
ic={x1[0]==-1,x2[0]==2,x3[0]==-30}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to -e^t \\ \text {x2}(t)\to 2 e^t (2 t+1) \\ \text {x3}(t)\to 3 e^t \left (-8 t+e^t-11\right ) \\ \end{align*}

✓ Sympy. Time used: 0.125 (sec). Leaf size: 60

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
x__3 = Function("x__3") 
ode=[Eq(-x__1(t) + Derivative(x__1(t), t),0),Eq(4*x__1(t) - x__2(t) + Derivative(x__2(t), t),0),Eq(-3*x__1(t) - 6*x__2(t) - 2*x__3(t) + Derivative(x__3(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t),x__3(t)],ics=ics)
 

\[ \left [ x^{1}{\left (t \right )} = \frac {2 C_{1} e^{t}}{7}, \ x^{2}{\left (t \right )} = - \frac {8 C_{1} t e^{t}}{7} + \left (C_{1} - \frac {8 C_{2}}{7}\right ) e^{t}, \ x^{3}{\left (t \right )} = \frac {48 C_{1} t e^{t}}{7} + \frac {48 C_{2} e^{t}}{7} + C_{3} e^{2 t}\right ] \]