9.5.3 problem Example 4

Internal problem ID [1006]
Book : Differential equations and linear algebra, 4th ed., Edwards and Penney
Section : Section 7.6, Multiple Eigenvalue Solutions. Examples. Page 437
Problem number : Example 4
Date solved : Tuesday, March 04, 2025 at 12:07:20 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.034 (sec). Leaf size: 74

ode:=[diff(x__1(t),t) = x__2(t)+2*x__3(t), diff(x__2(t),t) = -5*x__1(t)-3*x__2(t)-7*x__3(t), diff(x__3(t),t) = x__1(t)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 134

ode={D[ x1[t],t]==0*x1[t]+1*x2[t]+2*x3[t],D[ x2[t],t]==-5*x1[t]-3*x2[t]-7*x3[t],D[ x3[t],t]==1*x1[t]+0*x2[t]+0*x[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to \frac {1}{2} e^{-t} \left (c_1 \left (-2 t^2+2 t+2\right )-c_2 (t-2) t+c_3 (4-3 t) t\right ) \\ \text {x2}(t)\to \frac {1}{2} e^{-t} \left (-\left ((2 c_1+c_2+3 c_3) t^2\right )-2 (5 c_1+2 c_2+7 c_3) t+2 c_2\right ) \\ \text {x3}(t)\to \frac {1}{2} e^{-t} \left ((2 c_1+c_2+3 c_3) t^2+2 (c_1+c_3) t+2 c_3\right ) \\ \end{align*}

✓ Sympy. Time used: 0.155 (sec). Leaf size: 94

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
x__3 = Function("x__3") 
ode=[Eq(-x__2(t) - 2*x__3(t) + Derivative(x__1(t), t),0),Eq(5*x__1(t) + 3*x__2(t) + 7*x__3(t) + Derivative(x__2(t), t),0),Eq(-x__1(t) + Derivative(x__3(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t),x__3(t)],ics=ics)
 

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