20.18.8 problem 8

Internal problem ID [3878]
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 : 8
Date solved : Tuesday, March 04, 2025 at 05:18:32 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.093 (sec). Leaf size: 71

ode:=[diff(x__1(t),t) = x__1(t)-exp(t), diff(x__2(t),t) = 2*x__1(t)-3*x__2(t)+2*x__3(t)+6*exp(-t), diff(x__3(t),t) = x__1(t)-2*x__2(t)+2*x__3(t)+exp(t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= \left (-t +c_3 \right ) {\mathrm e}^{t} \\ x_{2} \left (t \right ) &= \frac {c_{2} {\mathrm e}^{t}}{2}+\frac {c_3 \,{\mathrm e}^{t}}{2}-\frac {{\mathrm e}^{t}}{6}+9 \,{\mathrm e}^{-t}+2 c_{1} {\mathrm e}^{-2 t} \\ x_{3} \left (t \right ) &= c_{2} {\mathrm e}^{t}+c_{1} {\mathrm e}^{-2 t}+t \,{\mathrm e}^{t}-\frac {{\mathrm e}^{t}}{3}+6 \,{\mathrm e}^{-t} \\ \end{align*}

✓ Mathematica. Time used: 0.139 (sec). Leaf size: 113

ode={D[x1[t],t]==x1[t]-Exp[t],D[x2[t],t]==2*x1[t]-3*x2[t]+2*x3[t]+6*Exp[-t],D[x3[t],t]==x1[t]-2*x2[t]+2*x3[t]+Exp[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to e^t (-t+c_1) \\ \text {x2}(t)\to \frac {1}{3} e^{-2 t} \left (27 e^t+(2 c_1-c_2+2 c_3) e^{3 t}-2 (c_1-2 c_2+c_3)\right ) \\ \text {x3}(t)\to \frac {1}{3} e^{-2 t} \left (18 e^t+e^{3 t} (3 t+c_1-2 c_2+4 c_3)-c_1+2 c_2-c_3\right ) \\ \end{align*}

✓ Sympy. Time used: 0.229 (sec). Leaf size: 61

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) + exp(t) + Derivative(x__1(t), t),0),Eq(-2*x__1(t) + 3*x__2(t) - 2*x__3(t) + Derivative(x__2(t), t) - 6*exp(-t),0),Eq(-x__1(t) + 2*x__2(t) - 2*x__3(t) - exp(t) + Derivative(x__3(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t),x__3(t)],ics=ics)
 

\[ \left [ x^{1}{\left (t \right )} = - t e^{t} - \left (C_{1} - 2 C_{2}\right ) e^{t}, \ x^{2}{\left (t \right )} = C_{2} e^{t} + 2 C_{3} e^{- 2 t} + 9 e^{- t}, \ x^{3}{\left (t \right )} = C_{1} e^{t} + C_{3} e^{- 2 t} + t e^{t} + 6 e^{- t}\right ] \]