76.25.19 problem 23

Internal problem ID [17761]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 6. Systems of First Order Linear Equations. Section 6.3 (Homogeneous Linear Systems with Constant Coefficients). Problems at page 408
Problem number : 23
Date solved : Monday, March 31, 2025 at 04:26:53 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=-3 x_{2} \left (t \right )-2 x_{3} \left (t \right )+3 x_{4} \left (t \right )+2 x_{5} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=8 x_{1} \left (t \right )+6 x_{2} \left (t \right )+4 x_{3} \left (t \right )-8 x_{4} \left (t \right )-16 x_{5} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=-8 x_{1} \left (t \right )-8 x_{2} \left (t \right )-6 x_{3} \left (t \right )+8 x_{4} \left (t \right )-16 x_{5} \left (t \right )\\ \frac {d}{d t}x_{4} \left (t \right )&=8 x_{1} \left (t \right )+7 x_{2} \left (t \right )+4 x_{3} \left (t \right )-9 x_{4} \left (t \right )-16 x_{5} \left (t \right )\\ \frac {d}{d t}x_{5} \left (t \right )&=-3 x_{1} \left (t \right )-5 x_{2} \left (t \right )-3 x_{3} \left (t \right )+5 x_{4} \left (t \right )+7 x_{5} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.204 (sec). Leaf size: 396

ode:=[diff(x__1(t),t) = -3*x__2(t)-2*x__3(t)+3*x__4(t)+2*x__5(t), diff(x__2(t),t) = 8*x__1(t)+6*x__2(t)+4*x__3(t)-8*x__4(t)-16*x__5(t), diff(x__3(t),t) = -8*x__1(t)-8*x__2(t)-6*x__3(t)+8*x__4(t)-16*x__5(t), diff(x__4(t),t) = 8*x__1(t)+7*x__2(t)+4*x__3(t)-9*x__4(t)-16*x__5(t), diff(x__5(t),t) = -3*x__1(t)-5*x__2(t)-3*x__3(t)+5*x__4(t)+7*x__5(t)]; 
dsolve(ode);
 

\begin{align*} \text {Solution too large to show}\end{align*}

✓ Mathematica. Time used: 0.015 (sec). Leaf size: 408

ode={D[x1[t],t]==0*x1[t]-3*x2[t]-2*x3[t]+3*x4[t]+2*x5[t],D[x2[t],t]==8*x1[t]+6*x2[t]+4*x3[t]-8*x4[t]-16*x5[t],D[x3[t],t]==-8*x1[t]-8*x2[t]-6*x3[t]+8*x4[t]+16*x5[t],D[x4[t],t]==8*x1[t]+7*x2[t]+4*x3[t]-9*x4[t]-16*x5[t],D[x5[t],t]==-3*x1[t]-5*x2[t]-3*x3[t]+5*x4[t]+7*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 (-2 e^{3 t}+2 e^{4 t}+1\right )+c_2 \left (e^t-4 e^{3 t}+2 e^{4 t}+1\right )-2 c_3 e^{3 t}+c_3 e^{4 t}-c_4 e^t+4 c_4 e^{3 t}-2 c_4 e^{4 t}+6 c_5 e^{3 t}-4 c_5 e^{4 t}+c_3-c_4-2 c_5\right ) \\ \text {x2}(t)\to e^{-2 t} \left (2 c_1 \left (e^{4 t}-1\right )+c_2 \left (2 e^{4 t}-1\right )+(c_3-2 c_4-4 c_5) \left (e^{4 t}-1\right )\right ) \\ \text {x3}(t)\to e^{-2 t} \left (-2 c_1 \left (e^{4 t}-1\right )-2 c_2 \left (e^{4 t}-1\right )-c_3 e^{4 t}+2 c_4 e^{4 t}+4 c_5 e^{4 t}+2 c_3-2 c_4-4 c_5\right ) \\ \text {x4}(t)\to e^{-2 t} \left (2 c_1 \left (e^{4 t}-1\right )+c_2 \left (-e^t+2 e^{4 t}-1\right )+c_3 e^{4 t}+c_4 e^t-2 c_4 e^{4 t}-4 c_5 e^{4 t}-c_3+2 c_4+4 c_5\right ) \\ \text {x5}(t)\to e^{-2 t} \left (-\left (c_1 \left (e^{3 t}-1\right )\right )+c_2 \left (e^t-2 e^{3 t}+1\right )-c_3 e^{3 t}-c_4 e^t+2 c_4 e^{3 t}+3 c_5 e^{3 t}+c_3-c_4-2 c_5\right ) \\ \end{align*}

✓ Sympy. Time used: 0.764 (sec). Leaf size: 296

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(3*x__2(t) + 2*x__3(t) - 3*x__4(t) - 2*x__5(t) + Derivative(x__1(t), t),0),Eq(-8*x__1(t) - 6*x__2(t) - 4*x__3(t) + 8*x__4(t) + 16*x__5(t) + Derivative(x__2(t), t),0),Eq(8*x__1(t) + 8*x__2(t) + 6*x__3(t) - 8*x__4(t) + 16*x__5(t) + Derivative(x__3(t), t),0),Eq(-8*x__1(t) - 7*x__2(t) - 4*x__3(t) + 9*x__4(t) + 16*x__5(t) + Derivative(x__4(t), t),0),Eq(3*x__1(t) + 5*x__2(t) + 3*x__3(t) - 5*x__4(t) - 7*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)
 

\[ \left [ x^{1}{\left (t \right )} = \frac {C_{1} \left (87 - \sqrt {393}\right ) e^{- \frac {t \left (1 - \sqrt {393}\right )}{2}}}{138} + \frac {C_{2} \left (\sqrt {393} + 87\right ) e^{- \frac {t \left (1 + \sqrt {393}\right )}{2}}}{138} - \frac {13 C_{3} e^{- t}}{3} + C_{4} e^{2 t}, \ x^{2}{\left (t \right )} = - \frac {C_{1} \left (\sqrt {393} + 189\right ) e^{- \frac {t \left (1 - \sqrt {393}\right )}{2}}}{138} - \frac {C_{2} \left (189 - \sqrt {393}\right ) e^{- \frac {t \left (1 + \sqrt {393}\right )}{2}}}{138} + \frac {128 C_{3} e^{- t}}{3} + C_{4} e^{2 t} + C_{5} e^{- 2 t}, \ x^{3}{\left (t \right )} = \frac {C_{1} \left (129 - 11 \sqrt {393}\right ) e^{- \frac {t \left (1 - \sqrt {393}\right )}{2}}}{69} + \frac {C_{2} \left (129 + 11 \sqrt {393}\right ) e^{- \frac {t \left (1 + \sqrt {393}\right )}{2}}}{69} - \frac {224 C_{3} e^{- t}}{3} - C_{4} e^{2 t}, \ x^{4}{\left (t \right )} = - \frac {C_{1} \left (\sqrt {393} + 189\right ) e^{- \frac {t \left (1 - \sqrt {393}\right )}{2}}}{138} - \frac {C_{2} \left (189 - \sqrt {393}\right ) e^{- \frac {t \left (1 + \sqrt {393}\right )}{2}}}{138} - \frac {19 C_{3} e^{- t}}{3} + C_{4} e^{2 t} + C_{5} e^{- 2 t}, \ x^{5}{\left (t \right )} = C_{1} e^{- \frac {t \left (1 - \sqrt {393}\right )}{2}} + C_{2} e^{- \frac {t \left (1 + \sqrt {393}\right )}{2}} + C_{3} e^{- t}\right ] \]