8.25.16 problem 18

Internal problem ID [2773]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Section 3.12, Systems of differential equations. The nonhomogeneous equation. variation of parameters. Page 366
Problem number : 18
Date solved : Tuesday, September 30, 2025 at 05:51:57 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.189 (sec). Leaf size: 86

ode:=[diff(x__1(t),t) = 3*x__1(t)+2*x__2(t)+4*x__3(t)+2*exp(8*t), diff(x__2(t),t) = 2*x__1(t)+2*x__3(t)+exp(8*t), diff(x__3(t),t) = 4*x__1(t)+2*x__2(t)+3*x__3(t)+2*exp(8*t)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 139

ode={D[ x1[t],t]==3*x1[t]+2*x2[t]+4*x3[t]+2*Exp[8*t],D[ x2[t],t]==2*x1[t]+0*x2[t]+2*x3[t]+Exp[8*t],D[ x3[t],t]==4*x1[t]+2*x2[t]+3*x3[t]+2*Exp[8*t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.116 (sec). Leaf size: 70

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
x__3 = Function("x__3") 
ode=[Eq(-3*x__1(t) - 2*x__2(t) - 4*x__3(t) - 2*exp(8*t) + Derivative(x__1(t), t),0),Eq(-2*x__1(t) - 2*x__3(t) - exp(8*t) + Derivative(x__2(t), t),0),Eq(-4*x__1(t) - 2*x__2(t) - 3*x__3(t) - 2*exp(8*t) + Derivative(x__3(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t),x__3(t)],ics=ics)
 

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