9.4.34 problem problem 45

Internal problem ID [998]
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 45
Date solved : Tuesday, March 04, 2025 at 12:07:06 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=9 x_{1} \left (t \right )-7 x_{2} \left (t \right )-5 x_{3} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=-12 x_{1} \left (t \right )+7 x_{2} \left (t \right )+11 x_{3} \left (t \right )+9 x_{4} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=24 x_{1} \left (t \right )-17 x_{2} \left (t \right )-19 x_{3} \left (t \right )-9 x_{4} \left (t \right )\\ \frac {d}{d t}x_{4} \left (t \right )&=-18 x_{1} \left (t \right )+13 x_{2} \left (t \right )+17 x_{3} \left (t \right )+9 x_{4} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.039 (sec). Leaf size: 104

ode:=[diff(x__1(t),t) = 9*x__1(t)-7*x__2(t)-5*x__3(t), diff(x__2(t),t) = -12*x__1(t)+7*x__2(t)+11*x__3(t)+9*x__4(t), diff(x__3(t),t) = 24*x__1(t)-17*x__2(t)-19*x__3(t)-9*x__4(t), diff(x__4(t),t) = -18*x__1(t)+13*x__2(t)+17*x__3(t)+9*x__4(t)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 430

ode={D[ x1[t],t]==9*x1[t]-7*x2[t]-5*x3[t]+0*x4[t],D[ x2[t],t]==-12*x1[t]+7*x2[t]+11*x3[t]+9*x4[t],D[ x3[t],t]==24*x1[t]-17*x2[t]-19*x3[t]-9*x4[t],D[ x4[t],t]==-18*x1[t]+13*x2[t]+17*x3[t]+9*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}{3} e^{-3 t} \left (c_1 \left (6 e^{3 t}-6 e^{6 t}+6 e^{9 t}-3\right )-\left (e^{3 t}-1\right ) \left (c_2 \left (4 e^{6 t}+3\right )+c_3 \left (-3 e^{3 t}+5 e^{6 t}+3\right )+3 c_4 e^{3 t} \left (e^{3 t}-1\right )\right )\right ) \\ \text {x2}(t)\to \frac {1}{3} e^{-3 t} \left (-3 c_1 \left (-4 e^{3 t}+e^{6 t}+2 e^{9 t}+1\right )+c_2 \left (-6 e^{3 t}+2 e^{6 t}+4 e^{9 t}+3\right )+\left (e^{3 t}-1\right ) \left (c_3 \left (9 e^{3 t}+5 e^{6 t}-3\right )+3 c_4 e^{3 t} \left (e^{3 t}+2\right )\right )\right ) \\ \text {x3}(t)\to c_1 \left (-e^{-3 t}-e^{3 t}+4 e^{6 t}-2\right )+c_2 \left (e^{-3 t}+\frac {2 e^{3 t}}{3}-\frac {8 e^{6 t}}{3}+1\right )+c_3 e^{-3 t}+\frac {4}{3} c_3 e^{3 t}-\frac {10}{3} c_3 e^{6 t}+c_4 e^{3 t}-2 c_4 e^{6 t}+2 c_3+c_4 \\ \text {x4}(t)\to \frac {1}{3} \left (c_1 \left (3 e^{-3 t}-3 e^{3 t}-6 e^{6 t}+6\right )+c_2 \left (-3 e^{-3 t}+2 e^{3 t}+4 e^{6 t}-3\right )-3 c_3 e^{-3 t}+4 c_3 e^{3 t}+5 c_3 e^{6 t}+3 c_4 e^{3 t}+3 c_4 e^{6 t}-6 c_3-3 c_4\right ) \\ \end{align*}

✓ Sympy. Time used: 0.216 (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(-9*x__1(t) + 7*x__2(t) + 5*x__3(t) + Derivative(x__1(t), t),0),Eq(12*x__1(t) - 7*x__2(t) - 11*x__3(t) - 9*x__4(t) + Derivative(x__2(t), t),0),Eq(-24*x__1(t) + 17*x__2(t) + 19*x__3(t) + 9*x__4(t) + Derivative(x__3(t), t),0),Eq(18*x__1(t) - 13*x__2(t) - 17*x__3(t) - 9*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)
 

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