20.18.9 problem 9

Internal problem ID [3879]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 9, First order linear systems. Section 9.6 (Variation of parameters for linear systems), page 624
Problem number : 9
Date solved : Tuesday, March 04, 2025 at 05:18:33 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.068 (sec). Leaf size: 65

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

\begin{align*} x_{1} \left (t \right ) &= \left (\frac {c_{1}}{3}-\frac {t}{3}+\frac {1}{9}\right ) {\mathrm e}^{3 t}+c_{2} \\ x_{2} \left (t \right ) &= -\frac {2 c_{1} {\mathrm e}^{3 t}}{3}+\frac {11 \,{\mathrm e}^{3 t} t}{3}-\frac {5 \,{\mathrm e}^{3 t}}{9}-\frac {c_{2}}{2}+c_3 \,{\mathrm e}^{3 t} \\ x_{3} \left (t \right ) &= \left (3 t +c_3 \right ) {\mathrm e}^{3 t} \\ \end{align*}

✓ Mathematica. Time used: 0.013 (sec). Leaf size: 106

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

\begin{align*} \text {x1}(t)\to \frac {1}{9} \left (6 (2 c_1+c_2-c_3)-e^{3 t} (3 t+2+3 c_1+6 c_2-6 c_3)\right ) \\ \text {x2}(t)\to \frac {1}{9} \left (e^{3 t} (33 t+1+6 c_1+12 c_2-3 c_3)-3 (2 c_1+c_2-c_3)\right ) \\ \text {x3}(t)\to e^{3 t} (3 t+c_3) \\ \end{align*}

✓ Sympy. Time used: 0.184 (sec). Leaf size: 71

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

\[ \left [ x^{1}{\left (t \right )} = - 2 C_{1} - \frac {t e^{3 t}}{3} - \left (\frac {C_{2}}{2} - \frac {C_{3}}{2} + \frac {2}{9}\right ) e^{3 t}, \ x^{2}{\left (t \right )} = C_{1} + \frac {11 t e^{3 t}}{3} + \left (C_{2} + \frac {1}{9}\right ) e^{3 t}, \ x^{3}{\left (t \right )} = C_{3} e^{3 t} + 3 t e^{3 t}\right ] \]