9.6.20 problem problem 20

Internal problem ID [1027]
Book : Differential equations and linear algebra, 4th ed., Edwards and Penney
Section : Section 7.6, Multiple Eigenvalue Solutions. Page 451
Problem number : problem 20
Date solved : Saturday, March 29, 2025 at 10:37:11 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.191 (sec). Leaf size: 77

ode:=[diff(x__1(t),t) = 2*x__1(t)+x__2(t)+x__4(t), diff(x__2(t),t) = 2*x__2(t)+x__3(t), diff(x__3(t),t) = 2*x__3(t)+x__4(t), diff(x__4(t),t) = 2*x__4(t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= \frac {\left (c_4 \,t^{3}+3 c_3 \,t^{2}+6 c_2 t +6 c_4 t +6 c_1 \right ) {\mathrm e}^{2 t}}{6} \\ x_{2} \left (t \right ) &= \frac {\left (c_4 \,t^{2}+2 c_3 t +2 c_2 \right ) {\mathrm e}^{2 t}}{2} \\ x_{3} \left (t \right ) &= \left (c_4 t +c_3 \right ) {\mathrm e}^{2 t} \\ x_{4} \left (t \right ) &= c_4 \,{\mathrm e}^{2 t} \\ \end{align*}

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 96

ode={D[ x1[t],t]==2*x1[t]+1*x2[t]+0*x3[t]+1*x4[t],D[ x2[t],t]==0*x1[t]+2*x2[t]+1*x3[t]+0*x4[t],D[ x3[t],t]==0*x1[t]+0*x2[t]+2*x3[t]+1*x4[t],D[ x4[t],t]==0*x1[t]+0*x2[t]+0*x3[t]+2*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}{6} e^{2 t} \left (t \left (c_4 t^2+3 c_3 t+6 c_2+6 c_4\right )+6 c_1\right ) \\ \text {x2}(t)\to \frac {1}{2} e^{2 t} (t (c_4 t+2 c_3)+2 c_2) \\ \text {x3}(t)\to e^{2 t} (c_4 t+c_3) \\ \text {x4}(t)\to c_4 e^{2 t} \\ \end{align*}

✓ Sympy. Time used: 0.177 (sec). Leaf size: 99

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(-2*x__1(t) - x__2(t) - x__4(t) + Derivative(x__1(t), t),0),Eq(-2*x__2(t) - x__3(t) + Derivative(x__2(t), t),0),Eq(-2*x__3(t) - x__4(t) + Derivative(x__3(t), t),0),Eq(-2*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)
 

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