9.6.24 problem problem 24

Internal problem ID [1031]
Book : Differential equations and linear algebra, 4th ed., Edwards and Penney
Section : Section 7.6, Multiple Eigenvalue Solutions. Page 451
Problem number : problem 24
Date solved : Saturday, March 29, 2025 at 10:37:16 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=28 x_{1} \left (t \right )+50 x_{2} \left (t \right )+100 x_{3} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=15 x_{1} \left (t \right )+33 x_{2} \left (t \right )+60 x_{3} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=-15 x_{1} \left (t \right )-30 x_{2} \left (t \right )-57 x_{3} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.179 (sec). Leaf size: 66

ode:=[diff(x__1(t),t) = 28*x__1(t)+50*x__2(t)+100*x__3(t), diff(x__2(t),t) = 15*x__1(t)+33*x__2(t)+60*x__3(t), diff(x__3(t),t) = -15*x__1(t)-30*x__2(t)-57*x__3(t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= c_2 \,{\mathrm e}^{-2 t}+c_3 \,{\mathrm e}^{3 t} \\ x_{2} \left (t \right ) &= \frac {3 c_2 \,{\mathrm e}^{-2 t}}{5}+\frac {3 c_3 \,{\mathrm e}^{3 t}}{5}+{\mathrm e}^{3 t} c_1 \\ x_{3} \left (t \right ) &= -\frac {3 c_2 \,{\mathrm e}^{-2 t}}{5}-\frac {11 c_3 \,{\mathrm e}^{3 t}}{20}-\frac {{\mathrm e}^{3 t} c_1}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.042 (sec). Leaf size: 229

ode={D[ x1[t],t]==28*x1[t]+50*x2[t]+100*x3[t],D[ x2[t],t]==15*x1[t]+33*x2[t]+60*x3[t],D[ x3[t],t]==-15*x1[t]-40*x2[t]-57*x3[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to \frac {1}{57} e^{t/2} \left (19 (3 c_1-5 c_2) e^{5 t/2}+95 c_2 \cos \left (\frac {5 \sqrt {95} t}{2}\right )+\sqrt {95} (6 c_1+13 c_2+24 c_3) \sin \left (\frac {5 \sqrt {95} t}{2}\right )\right ) \\ \text {x2}(t)\to \frac {1}{95} e^{t/2} \left (95 c_2 \cos \left (\frac {5 \sqrt {95} t}{2}\right )+\sqrt {95} (6 c_1+13 c_2+24 c_3) \sin \left (\frac {5 \sqrt {95} t}{2}\right )\right ) \\ \text {x3}(t)\to -\frac {e^{t/2} \left (95 (3 c_1-5 c_2) e^{5 t/2}-95 (3 c_1-5 c_2+12 c_3) \cos \left (\frac {5 \sqrt {95} t}{2}\right )+\sqrt {95} (69 c_1+197 c_2+276 c_3) \sin \left (\frac {5 \sqrt {95} t}{2}\right )\right )}{1140} \\ \end{align*}

✓ Sympy. Time used: 0.165 (sec). Leaf size: 56

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
x__3 = Function("x__3") 
ode=[Eq(-28*x__1(t) - 50*x__2(t) - 100*x__3(t) + Derivative(x__1(t), t),0),Eq(-15*x__1(t) - 33*x__2(t) - 60*x__3(t) + Derivative(x__2(t), t),0),Eq(15*x__1(t) + 30*x__2(t) + 57*x__3(t) + Derivative(x__3(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t),x__3(t)],ics=ics)
 

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