9.6.30 problem problem 30

Internal problem ID [1037]
Book : Differential equations and linear algebra, 4th ed., Edwards and Penney
Section : Section 7.6, Multiple Eigenvalue Solutions. Page 451
Problem number : problem 30
Date solved : Saturday, March 29, 2025 at 10:37:25 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=2 x_{1} \left (t \right )+x_{2} \left (t \right )-2 x_{3} \left (t \right )+x_{4} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=3 x_{2} \left (t \right )-5 x_{3} \left (t \right )+3 x_{4} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=-13 x_{2} \left (t \right )+22 x_{3} \left (t \right )-12 x_{4} \left (t \right )\\ \frac {d}{d t}x_{4} \left (t \right )&=-27 x_{2} \left (t \right )+45 x_{3} \left (t \right )-25 x_{4} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.192 (sec). Leaf size: 88

ode:=[diff(x__1(t),t) = 2*x__1(t)+x__2(t)-2*x__3(t)+x__4(t), diff(x__2(t),t) = 3*x__2(t)-5*x__3(t)+3*x__4(t), diff(x__3(t),t) = -13*x__2(t)+22*x__3(t)-12*x__4(t), diff(x__4(t),t) = -27*x__2(t)+45*x__3(t)-25*x__4(t)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 161

ode={D[ x1[t],t]==2*x1[t]+1*x2[t]-2*x3[t]+1*x4[t],D[ x2[t],t]==0*x1[t]+3*x2[t]-5*x3[t]+3*x4[t],D[ x3[t],t]==0*x1[t]-13*x2[t]+22*x3[t]-12*x4[t],D[ x4[t],t]==0*x1[t]-27*x2[t]+45*x3[t]-25*x4[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t],x4[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to e^{2 t} ((c_2-2 c_3+c_4) t+c_1) \\ \text {x2}(t)\to e^{-t} (4 c_2 t-5 c_3 t+3 c_4 t+c_2) \\ \text {x3}(t)\to e^{-t} \left (c_2 \left (-4 t-3 e^{3 t}+3\right )+c_3 \left (5 t+6 e^{3 t}-5\right )-3 c_4 \left (t+e^{3 t}-1\right )\right ) \\ \text {x4}(t)\to e^{-t} \left (c_2 \left (-12 t-5 e^{3 t}+5\right )+5 c_3 \left (3 t+2 e^{3 t}-2\right )-c_4 \left (9 t+5 e^{3 t}-6\right )\right ) \\ \end{align*}

✓ Sympy. Time used: 0.198 (sec). Leaf size: 95

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") 
ode=[Eq(-2*x__1(t) - x__2(t) + 2*x__3(t) - x__4(t) + Derivative(x__1(t), t),0),Eq(-3*x__2(t) + 5*x__3(t) - 3*x__4(t) + Derivative(x__2(t), t),0),Eq(13*x__2(t) - 22*x__3(t) + 12*x__4(t) + Derivative(x__3(t), t),0),Eq(27*x__2(t) - 45*x__3(t) + 25*x__4(t) + Derivative(x__4(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t),x__3(t),x__4(t)],ics=ics)
 

\[ \left [ x^{1}{\left (t \right )} = - \frac {C_{1} t e^{2 t}}{5} - \frac {C_{2} e^{2 t}}{5}, \ x^{2}{\left (t \right )} = 3 C_{3} t e^{- t} + \left (2 C_{3} + 3 C_{4}\right ) e^{- t}, \ x^{3}{\left (t \right )} = \frac {3 C_{1} e^{2 t}}{5} - 3 C_{3} t e^{- t} + \left (C_{3} - 3 C_{4}\right ) e^{- t}, \ x^{4}{\left (t \right )} = C_{1} e^{2 t} - 9 C_{3} t e^{- t} - 9 C_{4} e^{- t}\right ] \]