14.25.13 problem 14

Internal problem ID [2770]
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 : 14
Date solved : Tuesday, March 04, 2025 at 02:41:16 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 )+1\\ \frac {d}{d t}x_{2} \left (t \right )&=-4 x_{2} \left (t \right )-x_{3} \left (t \right )+t\\ \frac {d}{d t}x_{3} \left (t \right )&=5 x_{2} \left (t \right )+{\mathrm e}^{t} \end{align*}

✓ Maple. Time used: 0.211 (sec). Leaf size: 122

ode:=[diff(x__1(t),t) = -x__1(t)-x__2(t)+1, diff(x__2(t),t) = -4*x__2(t)-x__3(t)+t, diff(x__3(t),t) = 5*x__2(t)+exp(t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= -\frac {{\mathrm e}^{-2 t} \sin \left (t \right ) c_2}{2}+\frac {{\mathrm e}^{-2 t} \sin \left (t \right ) c_3}{2}+\frac {c_2 \,{\mathrm e}^{-2 t} \cos \left (t \right )}{2}+\frac {{\mathrm e}^{-2 t} \cos \left (t \right ) c_3}{2}+\frac {{\mathrm e}^{t}}{20}+\frac {4}{5}+{\mathrm e}^{-t} c_1 \\ x_{2} \left (t \right ) &= {\mathrm e}^{-2 t} \sin \left (t \right ) c_3 +c_2 \,{\mathrm e}^{-2 t} \cos \left (t \right )+\frac {1}{5}-\frac {{\mathrm e}^{t}}{10} \\ x_{3} \left (t \right ) &= -2 \,{\mathrm e}^{-2 t} \sin \left (t \right ) c_3 -{\mathrm e}^{-2 t} \cos \left (t \right ) c_3 -2 c_2 \,{\mathrm e}^{-2 t} \cos \left (t \right )+{\mathrm e}^{-2 t} \sin \left (t \right ) c_2 +\frac {{\mathrm e}^{t}}{2}-\frac {4}{5}+t \\ \end{align*}

✓ Mathematica. Time used: 0.884 (sec). Leaf size: 144

ode={D[ x1[t],t]==-1*x1[t]-1*x2[t]+0*x3[t]+1,D[ x2[t],t]==0*x1[t]-4*x2[t]-1*x3[t]+t,D[ x3[t],t]==0*x1[t]+5*x2[t]-0*x3[t]+Exp[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to \frac {1}{20} e^{-2 t} \left (e^t \left (16 e^t+e^{2 t}+10 (2 c_1+c_2+c_3)\right )-10 (c_2+c_3) \cos (t)-10 (3 c_2+c_3) \sin (t)\right ) \\ \text {x2}(t)\to \frac {1}{10} \left (2-e^t\right )+c_2 e^{-2 t} \cos (t)-(2 c_2+c_3) e^{-2 t} \sin (t) \\ \text {x3}(t)\to t+\frac {e^t}{2}+c_3 e^{-2 t} \cos (t)+(5 c_2+2 c_3) e^{-2 t} \sin (t)-\frac {4}{5} \\ \end{align*}

✓ Sympy. Time used: 1.069 (sec). Leaf size: 250

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) + Derivative(x__1(t), t) - 1,0),Eq(-t + 4*x__2(t) + x__3(t) + Derivative(x__2(t), t),0),Eq(-5*x__2(t) - exp(t) + Derivative(x__3(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t),x__3(t)],ics=ics)
 

\[ \left [ x^{1}{\left (t \right )} = C_{1} e^{- t} - \frac {t \sin ^{2}{\left (t \right )}}{2} - \frac {t \cos ^{2}{\left (t \right )}}{2} + \frac {t}{2} + \left (\frac {C_{2}}{10} + \frac {3 C_{3}}{10}\right ) e^{- 2 t} \sin {\left (t \right )} - \left (\frac {3 C_{2}}{10} - \frac {C_{3}}{10}\right ) e^{- 2 t} \cos {\left (t \right )} - \frac {e^{t} \sin ^{2}{\left (t \right )}}{5} - \frac {e^{t} \cos ^{2}{\left (t \right )}}{5} + \frac {e^{t}}{4} + \frac {3 \sin ^{2}{\left (t \right )}}{10} + \frac {3 \cos ^{2}{\left (t \right )}}{10} + \frac {1}{2}, \ x^{2}{\left (t \right )} = - \left (\frac {C_{2}}{5} - \frac {2 C_{3}}{5}\right ) e^{- 2 t} \sin {\left (t \right )} - \left (\frac {2 C_{2}}{5} + \frac {C_{3}}{5}\right ) e^{- 2 t} \cos {\left (t \right )} - \frac {e^{t} \sin ^{2}{\left (t \right )}}{10} - \frac {e^{t} \cos ^{2}{\left (t \right )}}{10} + \frac {\sin ^{2}{\left (t \right )}}{5} + \frac {\cos ^{2}{\left (t \right )}}{5}, \ x^{3}{\left (t \right )} = C_{2} e^{- 2 t} \cos {\left (t \right )} - C_{3} e^{- 2 t} \sin {\left (t \right )} + t \sin ^{2}{\left (t \right )} + t \cos ^{2}{\left (t \right )} + \frac {e^{t} \sin ^{2}{\left (t \right )}}{2} + \frac {e^{t} \cos ^{2}{\left (t \right )}}{2} - \frac {4 \sin ^{2}{\left (t \right )}}{5} - \frac {4 \cos ^{2}{\left (t \right )}}{5}\right ] \]