14.22.6 problem 6

Internal problem ID [2733]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Section 3.8, Systems of differential equations. The eigenva1ue-eigenvector method. Page 339
Problem number : 6
Date solved : Tuesday, March 04, 2025 at 02:40:31 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=x_{1} \left (t \right )+2 x_{2} \left (t \right )+3 x_{3} \left (t \right )+6 x_{4} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=3 x_{1} \left (t \right )+6 x_{2} \left (t \right )+9 x_{3} \left (t \right )+18 x_{4} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=5 x_{1} \left (t \right )+10 x_{2} \left (t \right )+15 x_{3} \left (t \right )+30 x_{4} \left (t \right )\\ \frac {d}{d t}x_{4} \left (t \right )&=7 x_{1} \left (t \right )+14 x_{2} \left (t \right )+21 x_{3} \left (t \right )+42 x_{4} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.044 (sec). Leaf size: 62

ode:=[diff(x__1(t),t) = x__1(t)+2*x__2(t)+3*x__3(t)+6*x__4(t), diff(x__2(t),t) = 3*x__1(t)+6*x__2(t)+9*x__3(t)+18*x__4(t), diff(x__3(t),t) = 5*x__1(t)+10*x__2(t)+15*x__3(t)+30*x__4(t), diff(x__4(t),t) = 7*x__1(t)+14*x__2(t)+21*x__3(t)+42*x__4(t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= c_3 +c_4 \,{\mathrm e}^{64 t} \\ x_{2} \left (t \right ) &= 3 c_3 +3 c_4 \,{\mathrm e}^{64 t}+c_2 \\ x_{3} \left (t \right ) &= 5 c_3 +5 c_4 \,{\mathrm e}^{64 t}+c_1 \\ x_{4} \left (t \right ) &= 7 c_4 \,{\mathrm e}^{64 t}-\frac {11 c_3}{3}-\frac {c_2}{3}-\frac {c_1}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.095 (sec). Leaf size: 554

ode={D[ x1[t],t]==1*x1[t]+2*x2[t]+3*x3[t]+6*x4[t],D[ x2[t],t]==3*x1[t]+6*x2[t]+9*x3[t]+19*x4[t],D[ x3[t],t]==5*x1[t]+10*x2[t]+15*x3[t]+30*x4[t],D[ x4[t],t]==7*x1[t]+14*x2[t]+21*x3[t]+42*x4[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t],x4[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to \frac {e^{-\sqrt {1038} t} \left (2076 (7 c_1-c_4) e^{\sqrt {1038} t}-\left (7 \sqrt {1038} c_1+14 \sqrt {1038} c_2+21 \sqrt {1038} c_3+10 \sqrt {1038} c_4-1038 c_4\right ) e^{32 t}+\left (7 \sqrt {1038} c_1+14 \sqrt {1038} c_2+21 \sqrt {1038} c_3+10 \sqrt {1038} c_4+1038 c_4\right ) e^{2 \left (16+\sqrt {1038}\right ) t}\right )}{14532} \\ \text {x2}(t)\to \frac {\left (7 \left (519+13 \sqrt {1038}\right ) c_1+14 \left (519+13 \sqrt {1038}\right ) c_2+273 \sqrt {1038} c_3+10899 c_3-389 \sqrt {1038} c_4-8304 c_4\right ) e^{-\left (\left (\sqrt {1038}-32\right ) t\right )}+\left (\left (3633-91 \sqrt {1038}\right ) c_1+\left (7266-182 \sqrt {1038}\right ) c_2-273 \sqrt {1038} c_3+10899 c_3+389 \sqrt {1038} c_4-8304 c_4\right ) e^{\left (32+\sqrt {1038}\right ) t}-1038 (7 c_1+21 c_3-16 c_4)}{14532} \\ \text {x3}(t)\to \frac {e^{-\sqrt {1038} t} \left (2076 (7 c_3-5 c_4) e^{\sqrt {1038} t}-5 \left (7 \sqrt {1038} c_1+14 \sqrt {1038} c_2+21 \sqrt {1038} c_3+10 \sqrt {1038} c_4-1038 c_4\right ) e^{32 t}+5 \left (7 \sqrt {1038} c_1+14 \sqrt {1038} c_2+21 \sqrt {1038} c_3+10 \sqrt {1038} c_4+1038 c_4\right ) e^{2 \left (16+\sqrt {1038}\right ) t}\right )}{14532} \\ \text {x4}(t)\to \frac {e^{-\left (\left (\sqrt {1038}-32\right ) t\right )} \left (7 c_1 \left (e^{2 \sqrt {1038} t}-1\right )+14 c_2 \left (e^{2 \sqrt {1038} t}-1\right )+21 c_3 e^{2 \sqrt {1038} t}+\sqrt {1038} c_4 e^{2 \sqrt {1038} t}+10 c_4 e^{2 \sqrt {1038} t}-21 c_3+\sqrt {1038} c_4-10 c_4\right )}{2 \sqrt {1038}} \\ \end{align*}

✓ Sympy. Time used: 0.180 (sec). Leaf size: 58

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) - 2*x__2(t) - 3*x__3(t) - 6*x__4(t) + Derivative(x__1(t), t),0),Eq(-3*x__1(t) - 6*x__2(t) - 9*x__3(t) - 18*x__4(t) + Derivative(x__2(t), t),0),Eq(-5*x__1(t) - 10*x__2(t) - 15*x__3(t) - 30*x__4(t) + Derivative(x__3(t), t),0),Eq(-7*x__1(t) - 14*x__2(t) - 21*x__3(t) - 42*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)
 

\[ \left [ x^{1}{\left (t \right )} = - 6 C_{1} - 3 C_{2} - 2 C_{3} + \frac {C_{4} e^{64 t}}{7}, \ x^{2}{\left (t \right )} = C_{3} + \frac {3 C_{4} e^{64 t}}{7}, \ x^{3}{\left (t \right )} = C_{2} + \frac {5 C_{4} e^{64 t}}{7}, \ x^{4}{\left (t \right )} = C_{1} + C_{4} e^{64 t}\right ] \]