9.6.22 problem problem 22

Internal problem ID [1029]
Book : Differential equations and linear algebra, 4th ed., Edwards and Penney
Section : Section 7.6, Multiple Eigenvalue Solutions. Page 451
Problem number : problem 22
Date solved : Saturday, March 29, 2025 at 10:37:13 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=x_{1} \left (t \right )+3 x_{2} \left (t \right )+7 x_{3} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=-x_{2} \left (t \right )-4 x_{3} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=x_{2} \left (t \right )+3 x_{3} \left (t \right )\\ \frac {d}{d t}x_{4} \left (t \right )&=-6 x_{2} \left (t \right )-14 x_{3} \left (t \right )+x_{4} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.184 (sec). Leaf size: 77

ode:=[diff(x__1(t),t) = x__1(t)+3*x__2(t)+7*x__3(t), diff(x__2(t),t) = -x__2(t)-4*x__3(t), diff(x__3(t),t) = x__2(t)+3*x__3(t), diff(x__4(t),t) = -6*x__2(t)-14*x__3(t)+x__4(t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= \frac {\left (-c_4 \,t^{2}-2 c_3 t -7 c_4 t +4 c_2 \right ) {\mathrm e}^{t}}{4} \\ x_{2} \left (t \right ) &= {\mathrm e}^{t} \left (c_4 t +c_3 \right ) \\ x_{3} \left (t \right ) &= -\frac {{\mathrm e}^{t} \left (2 c_4 t +2 c_3 +c_4 \right )}{4} \\ x_{4} \left (t \right ) &= \frac {\left (c_4 \,t^{2}+2 c_3 t +7 c_4 t +2 c_1 \right ) {\mathrm e}^{t}}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 99

ode={D[ x1[t],t]==1*x1[t]+3*x2[t]+7*x3[t]+0*x4[t],D[ x2[t],t]==0*x1[t]-1*x2[t]-4*x3[t]+0*x4[t],D[ x3[t],t]==0*x1[t]+1*x2[t]+3*x3[t]+0*x4[t],D[ x4[t],t]==0*x1[t]-6*x2[t]-14*x3[t]+1*x4[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t],x4[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to \frac {1}{2} e^t (c_2 t (t+6)+2 c_3 t (t+7)+2 c_1) \\ \text {x2}(t)\to -e^t (c_2 (2 t-1)+4 c_3 t) \\ \text {x3}(t)\to e^t ((c_2+2 c_3) t+c_3) \\ \text {x4}(t)\to e^t (c_2 (-t) (t+6)-2 c_3 t (t+7)+c_4) \\ \end{align*}

✓ Sympy. Time used: 0.180 (sec). Leaf size: 100

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
x__3 = Function("x__3") 
x__4 = Function("x__4") 
ode=[Eq(-x__1(t) - 3*x__2(t) - 7*x__3(t) + Derivative(x__1(t), t),0),Eq(x__2(t) + 4*x__3(t) + Derivative(x__2(t), t),0),Eq(-x__2(t) - 3*x__3(t) + Derivative(x__3(t), t),0),Eq(6*x__2(t) + 14*x__3(t) - x__4(t) + Derivative(x__4(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t),x__3(t),x__4(t)],ics=ics)
 

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