14.25.8 problem 6

Internal problem ID [2765]
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 : 6
Date solved : Sunday, March 30, 2025 at 12:16:49 AM
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 )-2 x_{3} \left (t \right )+{\mathrm e}^{t}\\ \frac {d}{d t}x_{2} \left (t \right )&=x_{1} \left (t \right )+x_{2} \left (t \right )+x_{3} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=2 x_{1} \left (t \right )+x_{2} \left (t \right )+3 x_{3} \left (t \right ) \end{align*}

With initial conditions

\begin{align*} x_{1} \left (0\right ) = 0\\ x_{2} \left (0\right ) = 0\\ x_{3} \left (0\right ) = 0 \end{align*}

✓ Maple. Time used: 0.227 (sec). Leaf size: 46

ode:=[diff(x__1(t),t) = -x__1(t)-x__2(t)-2*x__3(t)+exp(t), diff(x__2(t),t) = x__1(t)+x__2(t)+x__3(t), diff(x__3(t),t) = 2*x__1(t)+x__2(t)+3*x__3(t)]; 
ic:=x__1(0) = 0x__2(0) = 0x__3(0) = 0; 
dsolve([ode,ic]);
 

\begin{align*} x_{1} \left (t \right ) &= -\frac {{\mathrm e}^{t} \left (t^{3}+6 t^{2}-6 t \right )}{6} \\ x_{2} \left (t \right ) &= \frac {t^{2} {\mathrm e}^{t}}{2} \\ x_{3} \left (t \right ) &= \frac {\left (t^{3}+6 t^{2}\right ) {\mathrm e}^{t}}{6} \\ \end{align*}

✓ Mathematica. Time used: 0.008 (sec). Leaf size: 50

ode={D[ x1[t],t]==-1*x1[t]-1*x2[t]-2*x3[t]+Exp[t],D[ x2[t],t]==1*x1[t]+1*x2[t]+1*x3[t],D[ x3[t],t]==2*x1[t]+1*x2[t]+3*x3[t]}; 
ic={x1[0]==0,x2[0]==0,x3[0]==0}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to -\frac {1}{6} e^t t \left (t^2+6 t-6\right ) \\ \text {x2}(t)\to \frac {e^t t^2}{2} \\ \text {x3}(t)\to \frac {1}{6} e^t t^2 (t+6) \\ \end{align*}

✓ Sympy. Time used: 0.224 (sec). Leaf size: 109

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

\[ \left [ x^{1}{\left (t \right )} = - \frac {t^{3} e^{t}}{6} - t^{2} \left (\frac {C_{3}}{2} + 1\right ) e^{t} - t \left (C_{1} + 2 C_{3} - 1\right ) e^{t} - \left (2 C_{1} + C_{2} - C_{3}\right ) e^{t}, \ x^{2}{\left (t \right )} = C_{1} e^{t} + C_{3} t e^{t} + \frac {t^{2} e^{t}}{2}, \ x^{3}{\left (t \right )} = \frac {t^{3} e^{t}}{6} + t^{2} \left (\frac {C_{3}}{2} + 1\right ) e^{t} + t \left (C_{1} + 2 C_{3}\right ) e^{t} + \left (2 C_{1} + C_{2}\right ) e^{t}\right ] \]