10.17.12 problem 12

Internal problem ID [1427]
Book : Elementary differential equations and boundary value problems, 10th ed., Boyce and DiPrima
Section : Chapter 7.8, Repeated Eigenvalues. page 436
Problem number : 12
Date solved : Tuesday, March 04, 2025 at 12:35:42 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=-\frac {5 x_{1} \left (t \right )}{2}+x_{2} \left (t \right )+x_{3} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=x_{1} \left (t \right )-\frac {5 x_{2} \left (t \right )}{2}+x_{3} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=x_{1} \left (t \right )+x_{2} \left (t \right )-\frac {5 x_{3} \left (t \right )}{2} \end{align*}

With initial conditions

\begin{align*} x_{1} \left (0\right ) = 2\\ x_{2} \left (0\right ) = 3\\ x_{3} \left (0\right ) = -1 \end{align*}

✓ Maple. Time used: 0.040 (sec). Leaf size: 49

ode:=[diff(x__1(t),t) = -5/2*x__1(t)+x__2(t)+x__3(t), diff(x__2(t),t) = x__1(t)-5/2*x__2(t)+x__3(t), diff(x__3(t),t) = x__1(t)+x__2(t)-5/2*x__3(t)]; 
ic:=x__1(0) = 2x__2(0) = 3x__3(0) = -1; 
dsolve([ode,ic]);
 

\begin{align*} x_{1} \left (t \right ) &= \frac {4 \,{\mathrm e}^{-\frac {t}{2}}}{3}+\frac {2 \,{\mathrm e}^{-\frac {7 t}{2}}}{3} \\ x_{2} \left (t \right ) &= \frac {4 \,{\mathrm e}^{-\frac {t}{2}}}{3}+\frac {5 \,{\mathrm e}^{-\frac {7 t}{2}}}{3} \\ x_{3} \left (t \right ) &= \frac {4 \,{\mathrm e}^{-\frac {t}{2}}}{3}-\frac {7 \,{\mathrm e}^{-\frac {7 t}{2}}}{3} \\ \end{align*}

✓ Mathematica. Time used: 0.008 (sec). Leaf size: 71

ode={D[ x1[t],t]==-5/2*x1[t]+1*x2[t]+1*x3[t],D[ x2[t],t]==1*x1[t]-5/2*x2[t]+1*x3[t],D[ x3[t],t]==1*x1[t]+1*x2[t]-5/2*x3[t]}; 
ic={x1[0]==2,x2[0]==3,x3[0]==-1}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to \frac {2}{3} e^{-7 t/2} \left (2 e^{3 t}+1\right ) \\ \text {x2}(t)\to \frac {1}{3} e^{-7 t/2} \left (4 e^{3 t}+5\right ) \\ \text {x3}(t)\to \frac {1}{3} e^{-7 t/2} \left (4 e^{3 t}-7\right ) \\ \end{align*}

✓ Sympy. Time used: 0.142 (sec). Leaf size: 53

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
x__3 = Function("x__3") 
ode=[Eq(5*x__1(t)/2 - x__2(t) - x__3(t) + Derivative(x__1(t), t),0),Eq(-x__1(t) + 5*x__2(t)/2 - x__3(t) + Derivative(x__2(t), t),0),Eq(-x__1(t) - x__2(t) + 5*x__3(t)/2 + Derivative(x__3(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t),x__3(t)],ics=ics)
 

\[ \left [ x^{1}{\left (t \right )} = C_{3} e^{- \frac {t}{2}} - \left (C_{1} + C_{2}\right ) e^{- \frac {7 t}{2}}, \ x^{2}{\left (t \right )} = C_{1} e^{- \frac {7 t}{2}} + C_{3} e^{- \frac {t}{2}}, \ x^{3}{\left (t \right )} = C_{2} e^{- \frac {7 t}{2}} + C_{3} e^{- \frac {t}{2}}\right ] \]