9.4.31 problem problem 42

Internal problem ID [995]
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 42
Date solved : Tuesday, March 04, 2025 at 12:07:03 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=-40 x_{1} \left (t \right )-12 x_{2} \left (t \right )+54 x_{3} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=35 x_{1} \left (t \right )+13 x_{2} \left (t \right )-46 x_{3} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=-25 x_{1} \left (t \right )-7 x_{2} \left (t \right )+34 x_{3} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.029 (sec). Leaf size: 58

ode:=[diff(x__1(t),t) = -40*x__1(t)-12*x__2(t)+54*x__3(t), diff(x__2(t),t) = 35*x__1(t)+13*x__2(t)-46*x__3(t), diff(x__3(t),t) = -25*x__1(t)-7*x__2(t)+34*x__3(t)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 181

ode={D[ x1[t],t]==-40*x1[t]-12*x2[t]+54*x3[t],D[ x2[t],t]==35*x1[t]+13*x2[t]-46*x3[t],D[ x3[t],t]==-25*x1[t]-7*x2[t]+34*x3[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to c_1 \left (-5 e^{2 t}-6 e^{5 t}+12\right )-c_2 \left (e^{2 t}+2 e^{5 t}-3\right )+c_3 \left (7 e^{2 t}+8 e^{5 t}-15\right ) \\ \text {x2}(t)\to c_1 \left (-5 e^{2 t}+9 e^{5 t}-4\right )+c_2 \left (-e^{2 t}+3 e^{5 t}-1\right )+c_3 \left (7 e^{2 t}-12 e^{5 t}+5\right ) \\ \text {x3}(t)\to c_1 \left (-5 e^{2 t}-3 e^{5 t}+8\right )-c_2 \left (e^{2 t}+e^{5 t}-2\right )+c_3 \left (7 e^{2 t}+4 e^{5 t}-10\right ) \\ \end{align*}

✓ Sympy. Time used: 0.129 (sec). Leaf size: 60

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
x__3 = Function("x__3") 
ode=[Eq(40*x__1(t) + 12*x__2(t) - 54*x__3(t) + Derivative(x__1(t), t),0),Eq(-35*x__1(t) - 13*x__2(t) + 46*x__3(t) + Derivative(x__2(t), t),0),Eq(25*x__1(t) + 7*x__2(t) - 34*x__3(t) + Derivative(x__3(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t),x__3(t)],ics=ics)
 

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