14.25.15 problem 17

Internal problem ID [2772]
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 : 17
Date solved : Tuesday, March 04, 2025 at 02:41:19 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.049 (sec). Leaf size: 86

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

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

✓ Mathematica. Time used: 0.008 (sec). Leaf size: 142

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

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

✓ Sympy. Time used: 0.189 (sec). Leaf size: 83

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

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