3.6.32 problem problem 32

Internal problem ID [1039]
Book : Differential equations and linear algebra, 4th ed., Edwards and Penney
Section : Section 7.6, Multiple Eigenvalue Solutions. Page 451
Problem number : problem 32
Date solved : Tuesday, September 30, 2025 at 04:21:04 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=11 x_{1} \left (t \right )-x_{2} \left (t \right )+26 x_{3} \left (t \right )+6 x_{4} \left (t \right )-3 x_{5} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=3 x_{2} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=-9 x_{1} \left (t \right )-24 x_{3} \left (t \right )-6 x_{4} \left (t \right )+3 x_{5} \left (t \right )\\ \frac {d}{d t}x_{4} \left (t \right )&=3 x_{1} \left (t \right )+9 x_{3} \left (t \right )+5 x_{4} \left (t \right )-x_{5} \left (t \right )\\ \frac {d}{d t}x_{5} \left (t \right )&=-48 x_{1} \left (t \right )-3 x_{2} \left (t \right )-138 x_{3} \left (t \right )-30 x_{4} \left (t \right )+18 x_{5} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.212 (sec). Leaf size: 106

ode:=[diff(x__1(t),t) = 11*x__1(t)-x__2(t)+26*x__3(t)+6*x__4(t)-3*x__5(t), diff(x__2(t),t) = 3*x__2(t), diff(x__3(t),t) = -9*x__1(t)-24*x__3(t)-6*x__4(t)+3*x__5(t), diff(x__4(t),t) = 3*x__1(t)+9*x__3(t)+5*x__4(t)-x__5(t), diff(x__5(t),t) = -48*x__1(t)-3*x__2(t)-138*x__3(t)-30*x__4(t)+18*x__5(t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= \left (-\left (c_4 +c_5 \right ) {\mathrm e}^{t}+c_1 \right ) {\mathrm e}^{2 t} \\ x_{2} \left (t \right ) &= c_5 \,{\mathrm e}^{3 t} \\ x_{3} \left (t \right ) &= c_3 \,{\mathrm e}^{2 t}+c_4 \,{\mathrm e}^{3 t} \\ x_{4} \left (t \right ) &= -\frac {c_3 \,{\mathrm e}^{2 t}}{3}-\frac {c_4 \,{\mathrm e}^{3 t}}{3}+{\mathrm e}^{3 t} c_2 \\ x_{5} \left (t \right ) &= 8 c_3 \,{\mathrm e}^{2 t}+\frac {16 c_4 \,{\mathrm e}^{3 t}}{3}+2 \,{\mathrm e}^{3 t} c_2 -3 c_5 \,{\mathrm e}^{3 t}+3 \,{\mathrm e}^{2 t} c_1 \\ \end{align*}

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 211

ode={D[ x1[t],t]==11*x1[t]-1*x2[t]+26*x3[t]+6*x4[t]-3*x5[t],D[ x2[t],t]==0*x1[t]+3*x2[t],D[ x3[t],t]==-9*x1[t]+0*x2[t]-24*x3[t]-6*x4[t]+3*x5[t],D[ x4[t],t]==3*x1[t]+0*x2[t]+9*x3[t]+5*x4[t]-1*x5[t],D[ x5[t],t]==-48*x1[t]-3*x2[t]-138*x3[t]-30*x4[t]+18*x5[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t],x4[t],x5[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)&\to e^{2 t} \left (c_1 \left (9 e^t-8\right )-(c_2-26 c_3-6 c_4+3 c_5) \left (e^t-1\right )\right )\\ \text {x2}(t)&\to c_2 e^{3 t}\\ \text {x3}(t)&\to -e^{2 t} \left (9 c_1 \left (e^t-1\right )+c_3 \left (26 e^t-27\right )+3 (2 c_4-c_5) \left (e^t-1\right )\right )\\ \text {x4}(t)&\to e^{2 t} \left (3 c_1 \left (e^t-1\right )+9 c_3 \left (e^t-1\right )+3 c_4 e^t-c_5 e^t-2 c_4+c_5\right )\\ \text {x5}(t)&\to -e^{2 t} \left (48 c_1 \left (e^t-1\right )+3 c_2 \left (e^t-1\right )+138 c_3 e^t+30 c_4 e^t-16 c_5 e^t-138 c_3-30 c_4+15 c_5\right ) \end{align*}

✓ Sympy. Time used: 0.213 (sec). Leaf size: 97

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") 
x__5 = Function("x__5") 
ode=[Eq(-11*x__1(t) + x__2(t) - 26*x__3(t) - 6*x__4(t) + 3*x__5(t) + Derivative(x__1(t), t),0),Eq(-3*x__2(t) + Derivative(x__2(t), t),0),Eq(9*x__1(t) + 24*x__3(t) + 6*x__4(t) - 3*x__5(t) + Derivative(x__3(t), t),0),Eq(-3*x__1(t) - 9*x__3(t) - 5*x__4(t) + x__5(t) + Derivative(x__4(t), t),0),Eq(48*x__1(t) + 3*x__2(t) + 138*x__3(t) + 30*x__4(t) - 18*x__5(t) + Derivative(x__5(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t),x__3(t),x__4(t),x__5(t)],ics=ics)
 

\[ \left [ x^{1}{\left (t \right )} = \left (\frac {C_{1}}{3} + 8 C_{2}\right ) e^{2 t} - \left (3 C_{3} + \frac {2 C_{4}}{3} - \frac {C_{5}}{3}\right ) e^{3 t}, \ x^{2}{\left (t \right )} = \left (2 C_{3} + \frac {2 C_{4}}{3} - \frac {C_{5}}{3}\right ) e^{3 t}, \ x^{3}{\left (t \right )} = - 3 C_{2} e^{2 t} + C_{3} e^{3 t}, \ x^{4}{\left (t \right )} = C_{2} e^{2 t} + C_{4} e^{3 t}, \ x^{5}{\left (t \right )} = C_{1} e^{2 t} + C_{5} e^{3 t}\right ] \]