14.24.8 problem 6

Internal problem ID [2755]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Section 3.10, Systems of differential equations. Equal roots. Page 352
Problem number : 6
Date solved : Sunday, March 30, 2025 at 12:16:31 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=-4 x_{1} \left (t \right )-4 x_{2} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=10 x_{1} \left (t \right )+9 x_{2} \left (t \right )+x_{3} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=-4 x_{1} \left (t \right )-3 x_{2} \left (t \right )+x_{3} \left (t \right ) \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.134 (sec). Leaf size: 57

ode:=[diff(x__1(t),t) = -4*x__1(t)-4*x__2(t), diff(x__2(t),t) = 10*x__1(t)+9*x__2(t)+x__3(t), diff(x__3(t),t) = -4*x__1(t)-3*x__2(t)+x__3(t)]; 
ic:=x__1(0) = 2x__2(0) = 1x__3(0) = -1; 
dsolve([ode,ic]);
 

\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{2 t} \left (-4 t^{2}-16 t +2\right ) \\ x_{2} \left (t \right ) &= -\frac {{\mathrm e}^{2 t} \left (-24 t^{2}-104 t -4\right )}{4} \\ x_{3} \left (t \right ) &= \frac {{\mathrm e}^{2 t} \left (-8 t^{2}-40 t -4\right )}{4} \\ \end{align*}

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 61

ode={D[ x1[t],t]==-4*x1[t]-4*x2[t]+0*x3[t],D[ x2[t],t]==10*x1[t]+9*x2[t]+1*x3[t],D[ x3[t],t]==-4*x1[t]-3*x2[t]+1*x3[t]}; 
ic={x1[0]==2,x2[0]==1,x3[0]==-1}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to -2 e^{2 t} \left (2 t^2+8 t-1\right ) \\ \text {x2}(t)\to e^{2 t} \left (6 t^2+26 t+1\right ) \\ \text {x3}(t)\to -e^{2 t} \left (2 t^2+10 t+1\right ) \\ \end{align*}

✓ Sympy. Time used: 0.196 (sec). Leaf size: 121

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
x__3 = Function("x__3") 
ode=[Eq(4*x__1(t) + 4*x__2(t) + Derivative(x__1(t), t),0),Eq(-10*x__1(t) - 9*x__2(t) - x__3(t) + Derivative(x__2(t), t),0),Eq(4*x__1(t) + 3*x__2(t) - x__3(t) + Derivative(x__3(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t),x__3(t)],ics=ics)
 

\[ \left [ x^{1}{\left (t \right )} = - 2 C_{3} t^{2} e^{2 t} - t \left (4 C_{1} + 6 C_{3}\right ) e^{2 t} - \left (6 C_{1} + 4 C_{2} - C_{3}\right ) e^{2 t}, \ x^{2}{\left (t \right )} = 3 C_{3} t^{2} e^{2 t} + t \left (6 C_{1} + 10 C_{3}\right ) e^{2 t} + \left (10 C_{1} + 6 C_{2}\right ) e^{2 t}, \ x^{3}{\left (t \right )} = - C_{3} t^{2} e^{2 t} - t \left (2 C_{1} + 4 C_{3}\right ) e^{2 t} - \left (4 C_{1} + 2 C_{2}\right ) e^{2 t}\right ] \]