9.6.28 problem problem 28

Internal problem ID [1035]
Book : Differential equations and linear algebra, 4th ed., Edwards and Penney
Section : Section 7.6, Multiple Eigenvalue Solutions. Page 451
Problem number : problem 28
Date solved : Saturday, March 29, 2025 at 10:37:22 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=-15 x_{1} \left (t \right )-7 x_{2} \left (t \right )+4 x_{3} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=34 x_{1} \left (t \right )+16 x_{2} \left (t \right )-11 x_{3} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=17 x_{1} \left (t \right )+7 x_{2} \left (t \right )+5 x_{3} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.116 (sec). Leaf size: 72

ode:=[diff(x__1(t),t) = -15*x__1(t)-7*x__2(t)+4*x__3(t), diff(x__2(t),t) = 34*x__1(t)+16*x__2(t)-11*x__3(t), diff(x__3(t),t) = 17*x__1(t)+7*x__2(t)+5*x__3(t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{2 t} \left (c_3 \,t^{2}+c_2 t +c_1 \right ) \\ x_{2} \left (t \right ) &= -\frac {{\mathrm e}^{2 t} \left (833 c_3 \,t^{2}+833 c_2 t +42 c_3 t +833 c_1 +21 c_2 -8 c_3 \right )}{343} \\ x_{3} \left (t \right ) &= \frac {{\mathrm e}^{2 t} \left (14 c_3 t +7 c_2 +2 c_3 \right )}{49} \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 124

ode={D[ x1[t],t]==-15*x1[t]-7*x2[t]+4*x3[t],D[ x2[t],t]==34*x1[t]+16*x2[t]-11*x3[t],D[ x3[t],t]==17*x1[t]+7*x2[t]+5*x3[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to \frac {1}{2} e^{2 t} \left (c_1 \left (119 t^2-34 t+2\right )+7 c_2 t (7 t-2)+c_3 t (21 t+8)\right ) \\ \text {x2}(t)\to -\frac {1}{2} e^{2 t} \left (17 (17 c_1+7 c_2+3 c_3) t^2+(-68 c_1-28 c_2+22 c_3) t-2 c_2\right ) \\ \text {x3}(t)\to e^{2 t} ((17 c_1+7 c_2+3 c_3) t+c_3) \\ \end{align*}

✓ Sympy. Time used: 0.181 (sec). Leaf size: 104

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
x__3 = Function("x__3") 
ode=[Eq(15*x__1(t) + 7*x__2(t) - 4*x__3(t) + Derivative(x__1(t), t),0),Eq(-34*x__1(t) - 16*x__2(t) + 11*x__3(t) + Derivative(x__2(t), t),0),Eq(-17*x__1(t) - 7*x__2(t) - 5*x__3(t) + Derivative(x__3(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t),x__3(t)],ics=ics)
 

\[ \left [ x^{1}{\left (t \right )} = \frac {119 C_{2} t^{2} e^{2 t}}{2} + t \left (119 C_{1} - 17 C_{2}\right ) e^{2 t} + \left (- 17 C_{1} + C_{2} + 119 C_{3}\right ) e^{2 t}, \ x^{2}{\left (t \right )} = - \frac {289 C_{2} t^{2} e^{2 t}}{2} - t \left (289 C_{1} - 34 C_{2}\right ) e^{2 t} + \left (34 C_{1} - 289 C_{3}\right ) e^{2 t}, \ x^{3}{\left (t \right )} = 17 C_{1} e^{2 t} + 17 C_{2} t e^{2 t}\right ] \]