3.4.32 problem problem 43

Internal problem ID [996]
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 43
Date solved : Tuesday, September 30, 2025 at 04:20:25 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=-20 x_{1} \left (t \right )+11 x_{2} \left (t \right )+13 x_{3} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=12 x_{1} \left (t \right )-x_{2} \left (t \right )-7 x_{3} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=-48 x_{1} \left (t \right )+21 x_{2} \left (t \right )+31 x_{3} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.123 (sec). Leaf size: 71

ode:=[diff(x__1(t),t) = -20*x__1(t)+11*x__2(t)+13*x__3(t), diff(x__2(t),t) = 12*x__1(t)-x__2(t)-7*x__3(t), diff(x__3(t),t) = -48*x__1(t)+21*x__2(t)+31*x__3(t)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.014 (sec). Leaf size: 554

ode={D[ x1[t],t]==20*x1[t]+11*x2[t]+13*x3[t],D[ x2[t],t]==12*x1[t]-1*x2[t]-7*x3[t],D[ x3[t],t]==-48*x1[t]+21*x2[t]+31*x3[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)&\to c_2 \text {RootSum}\left [\text {$\#$1}^3-50 \text {$\#$1}^2+1208 \text {$\#$1}-4576\&,\frac {11 \text {$\#$1} e^{\text {$\#$1} t}-68 e^{\text {$\#$1} t}}{3 \text {$\#$1}^2-100 \text {$\#$1}+1208}\&\right ]+c_3 \text {RootSum}\left [\text {$\#$1}^3-50 \text {$\#$1}^2+1208 \text {$\#$1}-4576\&,\frac {13 \text {$\#$1} e^{\text {$\#$1} t}-64 e^{\text {$\#$1} t}}{3 \text {$\#$1}^2-100 \text {$\#$1}+1208}\&\right ]+c_1 \text {RootSum}\left [\text {$\#$1}^3-50 \text {$\#$1}^2+1208 \text {$\#$1}-4576\&,\frac {\text {$\#$1}^2 e^{\text {$\#$1} t}-30 \text {$\#$1} e^{\text {$\#$1} t}+116 e^{\text {$\#$1} t}}{3 \text {$\#$1}^2-100 \text {$\#$1}+1208}\&\right ]\\ \text {x2}(t)&\to 12 c_1 \text {RootSum}\left [\text {$\#$1}^3-50 \text {$\#$1}^2+1208 \text {$\#$1}-4576\&,\frac {\text {$\#$1} e^{\text {$\#$1} t}-3 e^{\text {$\#$1} t}}{3 \text {$\#$1}^2-100 \text {$\#$1}+1208}\&\right ]-c_3 \text {RootSum}\left [\text {$\#$1}^3-50 \text {$\#$1}^2+1208 \text {$\#$1}-4576\&,\frac {7 \text {$\#$1} e^{\text {$\#$1} t}-296 e^{\text {$\#$1} t}}{3 \text {$\#$1}^2-100 \text {$\#$1}+1208}\&\right ]+c_2 \text {RootSum}\left [\text {$\#$1}^3-50 \text {$\#$1}^2+1208 \text {$\#$1}-4576\&,\frac {\text {$\#$1}^2 e^{\text {$\#$1} t}-51 \text {$\#$1} e^{\text {$\#$1} t}+1244 e^{\text {$\#$1} t}}{3 \text {$\#$1}^2-100 \text {$\#$1}+1208}\&\right ]\\ \text {x3}(t)&\to -12 c_1 \text {RootSum}\left [\text {$\#$1}^3-50 \text {$\#$1}^2+1208 \text {$\#$1}-4576\&,\frac {4 \text {$\#$1} e^{\text {$\#$1} t}-17 e^{\text {$\#$1} t}}{3 \text {$\#$1}^2-100 \text {$\#$1}+1208}\&\right ]+3 c_2 \text {RootSum}\left [\text {$\#$1}^3-50 \text {$\#$1}^2+1208 \text {$\#$1}-4576\&,\frac {7 \text {$\#$1} e^{\text {$\#$1} t}-316 e^{\text {$\#$1} t}}{3 \text {$\#$1}^2-100 \text {$\#$1}+1208}\&\right ]+c_3 \text {RootSum}\left [\text {$\#$1}^3-50 \text {$\#$1}^2+1208 \text {$\#$1}-4576\&,\frac {\text {$\#$1}^2 e^{\text {$\#$1} t}-19 \text {$\#$1} e^{\text {$\#$1} t}-152 e^{\text {$\#$1} t}}{3 \text {$\#$1}^2-100 \text {$\#$1}+1208}\&\right ] \end{align*}

✓ Sympy. Time used: 0.126 (sec). Leaf size: 75

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
x__3 = Function("x__3") 
ode=[Eq(20*x__1(t) - 11*x__2(t) - 13*x__3(t) + Derivative(x__1(t), t),0),Eq(-12*x__1(t) + x__2(t) + 7*x__3(t) + Derivative(x__2(t), t),0),Eq(48*x__1(t) - 21*x__2(t) - 31*x__3(t) + Derivative(x__3(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t),x__3(t)],ics=ics)
 

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