14.25.11 problem 12

Internal problem ID [2768]
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 : 12
Date solved : Tuesday, March 04, 2025 at 02:41:13 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=x_{1} \left (t \right )+3 x_{2} \left (t \right )+2 x_{3} \left (t \right )+\sin \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=-x_{1} \left (t \right )+2 x_{2} \left (t \right )+x_{3} \left (t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=4 x_{1} \left (t \right )-x_{2} \left (t \right )-x_{3} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.174 (sec). Leaf size: 82

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

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

✓ Mathematica. Time used: 0.262 (sec). Leaf size: 211

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

\begin{align*} \text {x1}(t)\to \frac {1}{30} \left (-3 \sin (t)-6 \cos (t)+5 e^{-2 t} \left (c_1 \left (e^{3 t}+3 e^{5 t}+2\right )+c_2 \left (-4 e^{3 t}+6 e^{5 t}-2\right )+c_3 \left (-e^{3 t}+3 e^{5 t}-2\right )\right )\right ) \\ \text {x2}(t)\to \frac {1}{30} \left (9 \sin (t)+3 \cos (t)-10 e^{-2 t} \left (c_1 \left (e^{3 t}-1\right )-4 c_2 e^{3 t}-c_3 e^{3 t}+c_2+c_3\right )\right ) \\ \text {x3}(t)\to \frac {1}{10} \left (-8 \sin (t)-\cos (t)+5 e^{-2 t} \left (c_1 \left (e^{3 t}+e^{5 t}-2\right )+2 c_2 \left (-2 e^{3 t}+e^{5 t}+1\right )+c_3 \left (-e^{3 t}+e^{5 t}+2\right )\right )\right ) \\ \end{align*}

✓ Sympy. Time used: 0.409 (sec). Leaf size: 97

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

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