76.29.12 problem 12

Internal problem ID [17817]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 6. Systems of First Order Linear Equations. Section 6.7 (Defective Matrices). Problems at page 444
Problem number : 12
Date solved : Monday, March 31, 2025 at 04:33:17 PM
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 )&=-14 x_{1} \left (t \right )-5 x_{2} \left (t \right )+x_{3} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=15 x_{1} \left (t \right )+5 x_{2} \left (t \right )-2 x_{3} \left (t \right ) \end{align*}

With initial conditions

\begin{align*} x_{1} \left (0\right ) = 5\\ x_{2} \left (0\right ) = 5\\ x_{3} \left (0\right ) = -4 \end{align*}

✓ Maple. Time used: 0.137 (sec). Leaf size: 56

ode:=[diff(x__1(t),t) = x__1(t)+x__2(t), diff(x__2(t),t) = -14*x__1(t)-5*x__2(t)+x__3(t), diff(x__3(t),t) = 15*x__1(t)+5*x__2(t)-2*x__3(t)]; 
ic:=x__1(0) = 5x__2(0) = 5x__3(0) = -4; 
dsolve([ode,ic]);
 

\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{-2 t} \left (-\frac {29}{2} t^{2}+20 t +5\right ) \\ x_{2} \left (t \right ) &= -{\mathrm e}^{-2 t} \left (-\frac {87}{2} t^{2}+89 t -5\right ) \\ x_{3} \left (t \right ) &= {\mathrm e}^{-2 t} \left (-\frac {145}{2} t^{2}+100 t -4\right ) \\ \end{align*}

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 68

ode={D[x1[t],t]==1*x1[t]+1*x2[t]+0*x3[t],D[x2[t],t]==-14*x1[t]-5*x2[t]+1*x3[t],D[x3[t],t]==15*x1[t]+5*x2[t]-2*x3[t]}; 
ic={x1[0]==5,x2[0]==5,x3[0]==-4}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to \frac {1}{2} e^{-2 t} \left (-29 t^2+40 t+10\right ) \\ \text {x2}(t)\to \frac {1}{2} e^{-2 t} \left (87 t^2-178 t+10\right ) \\ \text {x3}(t)\to -\frac {1}{2} e^{-2 t} \left (145 t^2-200 t+8\right ) \\ \end{align*}

✓ Sympy. Time used: 0.206 (sec). Leaf size: 124

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) - x__2(t) + Derivative(x__1(t), t),0),Eq(14*x__1(t) + 5*x__2(t) - x__3(t) + Derivative(x__2(t), t),0),Eq(-15*x__1(t) - 5*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)
 

\[ \left [ x^{1}{\left (t \right )} = - \frac {5 C_{2} t^{2} e^{- 2 t}}{2} + t \left (3 C_{2} - 5 C_{3}\right ) e^{- 2 t} + \left (- 5 C_{1} + C_{2} + 3 C_{3}\right ) e^{- 2 t}, \ x^{2}{\left (t \right )} = \frac {15 C_{2} t^{2} e^{- 2 t}}{2} - t \left (14 C_{2} - 15 C_{3}\right ) e^{- 2 t} + \left (15 C_{1} - 14 C_{3}\right ) e^{- 2 t}, \ x^{3}{\left (t \right )} = - \frac {25 C_{2} t^{2} e^{- 2 t}}{2} + t \left (15 C_{2} - 25 C_{3}\right ) e^{- 2 t} - \left (25 C_{1} - 15 C_{3}\right ) e^{- 2 t}\right ] \]