14.20.26 problem 26

Internal problem ID [3916]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 9, First order linear systems. Section 9.11 (Chapter review), page 665
Problem number : 26
Date solved : Tuesday, September 30, 2025 at 06:59:01 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=9 x_{1} \left (t \right )-2 x_{2} \left (t \right )+9 t\\ \frac {d}{d t}x_{2} \left (t \right )&=5 x_{1} \left (t \right )-2 x_{2} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.136 (sec). Leaf size: 43

ode:=[diff(x__1(t),t) = 9*x__1(t)-2*x__2(t)+9*t, diff(x__2(t),t) = 5*x__1(t)-2*x__2(t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= {\mathrm e}^{8 t} c_2 +{\mathrm e}^{-t} c_1 -\frac {9 t}{4}+\frac {27}{32} \\ x_{2} \left (t \right ) &= \frac {{\mathrm e}^{8 t} c_2}{2}+5 \,{\mathrm e}^{-t} c_1 +\frac {315}{64}-\frac {45 t}{8} \\ \end{align*}

✓ Mathematica. Time used: 0.164 (sec). Leaf size: 94

ode={D[x1[t],t]==9*x1[t]-2*x2[t]+9*t,D[x2[t],t]==5*x1[t]-2*x2[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)&\to -\frac {9 t}{4}-\frac {1}{9} (c_1-2 c_2) e^{-t}+\frac {2}{9} (5 c_1-c_2) e^{8 t}+\frac {27}{32}\\ \text {x2}(t)&\to -\frac {45 t}{8}-\frac {5}{9} (c_1-2 c_2) e^{-t}+\frac {1}{9} (5 c_1-c_2) e^{8 t}+\frac {315}{64} \end{align*}

✓ Sympy. Time used: 0.108 (sec). Leaf size: 48

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
ode=[Eq(-9*t - 9*x__1(t) + 2*x__2(t) + Derivative(x__1(t), t),0),Eq(-5*x__1(t) + 2*x__2(t) + Derivative(x__2(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
 

\[ \left [ x^{1}{\left (t \right )} = \frac {C_{1} e^{- t}}{5} + 2 C_{2} e^{8 t} - \frac {9 t}{4} + \frac {27}{32}, \ x^{2}{\left (t \right )} = C_{1} e^{- t} + C_{2} e^{8 t} - \frac {45 t}{8} + \frac {315}{64}\right ] \]