14.25.14 problem 16

Internal problem ID [2771]
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 : 16
Date solved : Tuesday, March 04, 2025 at 02:41:18 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}^{2 t}\\ \frac {d}{d t}x_{2} \left (t \right )&=2 x_{1} \left (t \right )+3 x_{2} \left (t \right )-4 x_{3} \left (t \right )+2 \,{\mathrm e}^{2 t}\\ \frac {d}{d t}x_{3} \left (t \right )&=4 x_{1} \left (t \right )+x_{2} \left (t \right )-4 x_{3} \left (t \right )+{\mathrm e}^{2 t} \end{align*}

✓ Maple. Time used: 0.050 (sec). Leaf size: 83

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

\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{2 t} t +{\mathrm e}^{t} c_1 +c_2 \,{\mathrm e}^{-3 t}+{\mathrm e}^{2 t} c_3 \\ x_{2} \left (t \right ) &= 2 \,{\mathrm e}^{2 t} t +{\mathrm e}^{t} c_1 +7 c_2 \,{\mathrm e}^{-3 t}+2 \,{\mathrm e}^{2 t} c_3 \\ x_{3} \left (t \right ) &= {\mathrm e}^{2 t} t +{\mathrm e}^{t} c_1 +11 c_2 \,{\mathrm e}^{-3 t}+{\mathrm e}^{2 t} c_3 \\ \end{align*}

✓ Mathematica. Time used: 0.111 (sec). Leaf size: 2491

ode={D[ x1[t],t]==1*x1[t]+1*x2[t]-1*x3[t]+Exp[2*t],D[ x2[t],t]==2*x1[t]+3*x2[t]-4*x3[t]+2*Exp[2*t],D[ x3[t],t]==4*x1[t]-1*x2[t]-4*x3[t]+Exp[2*t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions->True]
 

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✓ Sympy. Time used: 0.218 (sec). Leaf size: 90

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(2*t) + Derivative(x__1(t), t),0),Eq(-2*x__1(t) - 3*x__2(t) + 4*x__3(t) - 2*exp(2*t) + Derivative(x__2(t), t),0),Eq(-4*x__1(t) - x__2(t) + 4*x__3(t) - exp(2*t) + Derivative(x__3(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t),x__3(t)],ics=ics)
 

\[ \left [ x^{1}{\left (t \right )} = C_{1} e^{2 t} + \frac {C_{2} e^{- 3 t}}{11} + C_{3} e^{t} + t e^{2 t}, \ x^{2}{\left (t \right )} = 2 C_{1} e^{2 t} + \frac {7 C_{2} e^{- 3 t}}{11} + C_{3} e^{t} + 2 t e^{2 t}, \ x^{3}{\left (t \right )} = C_{1} e^{2 t} + C_{2} e^{- 3 t} + C_{3} e^{t} + t e^{2 t}\right ] \]