9.4.30 problem problem 41

Internal problem ID [994]
Book : Differential equations and linear algebra, 4th ed., Edwards and Penney
Section : Section 7.3, The eigenvalue method for linear systems. Page 395
Problem number : problem 41
Date solved : Tuesday, March 04, 2025 at 12:07:01 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=4 x_{1} \left (t \right )+x_{2} \left (t \right )+x_{3} \left (t \right )+7 x_{4} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=x_{1} \left (t \right )+4 x_{2} \left (t \right )+10 x_{3} \left (t \right )+x_{4} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=x_{1} \left (t \right )+10 x_{2} \left (t \right )+4 x_{3} \left (t \right )+x_{4} \left (t \right )\\ \frac {d}{d t}x_{4} \left (t \right )&=7 x_{1} \left (t \right )+x_{2} \left (t \right )+x_{3} \left (t \right )+4 x_{4} \left (t \right ) \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.018 (sec). Leaf size: 61

ode:=[diff(x__1(t),t) = 4*x__1(t)+x__2(t)+x__3(t)+7*x__4(t), diff(x__2(t),t) = x__1(t)+4*x__2(t)+10*x__3(t)+x__4(t), diff(x__3(t),t) = x__1(t)+10*x__2(t)+4*x__3(t)+x__4(t), diff(x__4(t),t) = 7*x__1(t)+x__2(t)+x__3(t)+4*x__4(t)]; 
ic:=x__1(0) = 3x__2(0) = 1x__3(0) = 1x__4(0) = 3; 
dsolve([ode,ic]);
 

\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{15 t}+2 \,{\mathrm e}^{10 t} \\ x_{2} \left (t \right ) &= 2 \,{\mathrm e}^{15 t}-{\mathrm e}^{10 t} \\ x_{3} \left (t \right ) &= 2 \,{\mathrm e}^{15 t}-{\mathrm e}^{10 t} \\ x_{4} \left (t \right ) &= {\mathrm e}^{15 t}+2 \,{\mathrm e}^{10 t} \\ \end{align*}

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 70

ode={D[ x1[t],t]==4*x1[t]+1*x2[t]+1*x3[t]+7*x4[t],D[ x2[t],t]==1*x1[t]+4*x2[t]+10*x3[t]+1*x4[t],D[ x3[t],t]==1*x1[t]+10*x2[t]+4*x3[t]+1*x4[t],D[ x4[t],t]==7*x1[t]+1*x2[t]+1*x3[t]+4*x4[t]}; 
ic={x1[0]==3,x2[0]==1,x3[0]==1,x4[0]==3}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t],x4[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to e^{10 t} \left (e^{5 t}+2\right ) \\ \text {x2}(t)\to e^{10 t} \left (2 e^{5 t}-1\right ) \\ \text {x3}(t)\to e^{10 t} \left (2 e^{5 t}-1\right ) \\ \text {x4}(t)\to e^{10 t} \left (e^{5 t}+2\right ) \\ \end{align*}

✓ Sympy. Time used: 0.214 (sec). Leaf size: 95

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(-4*x__1(t) - x__2(t) - x__3(t) - 7*x__4(t) + Derivative(x__1(t), t),0),Eq(-x__1(t) - 4*x__2(t) - 10*x__3(t) - x__4(t) + Derivative(x__2(t), t),0),Eq(-x__1(t) - 10*x__2(t) - 4*x__3(t) - x__4(t) + Derivative(x__3(t), t),0),Eq(-7*x__1(t) - x__2(t) - x__3(t) - 4*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)
 

\[ \left [ x^{1}{\left (t \right )} = - C_{1} e^{- 3 t} + C_{2} e^{10 t} + C_{3} e^{15 t}, \ x^{2}{\left (t \right )} = - \frac {C_{2} e^{10 t}}{2} + 2 C_{3} e^{15 t} - C_{4} e^{- 6 t}, \ x^{3}{\left (t \right )} = - \frac {C_{2} e^{10 t}}{2} + 2 C_{3} e^{15 t} + C_{4} e^{- 6 t}, \ x^{4}{\left (t \right )} = C_{1} e^{- 3 t} + C_{2} e^{10 t} + C_{3} e^{15 t}\right ] \]