14.20.27 problem 27

Internal problem ID [3917]
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 : 27
Date solved : Tuesday, September 30, 2025 at 06:59:02 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=10 x_{1} \left (t \right )-4 x_{2} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=4 x_{1} \left (t \right )+2 x_{2} \left (t \right )+\frac {{\mathrm e}^{6 t}}{t} \end{align*}

✓ Maple. Time used: 0.195 (sec). Leaf size: 56

ode:=[diff(x__1(t),t) = 10*x__1(t)-4*x__2(t), diff(x__2(t),t) = 4*x__1(t)+2*x__2(t)+1/t*exp(6*t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= -{\mathrm e}^{6 t} \left (4 t \ln \left (t \right )-t c_1 -c_2 -4 t \right ) \\ x_{2} \left (t \right ) &= -\frac {{\mathrm e}^{6 t} \left (16 t \ln \left (t \right )-4 t c_1 -4 \ln \left (t \right )+c_1 -4 c_2 -16 t \right )}{4} \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 61

ode={D[x1[t],t]==10*x1[t]-4*x2[t],D[x2[t],t]==4*x1[t]+2*x2[t]+1/t*Exp[6*t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)&\to e^{6 t} (-4 t \log (t)+4 (1+c_1-c_2) t+c_1)\\ \text {x2}(t)&\to e^{6 t} ((1-4 t) \log (t)+4 (1+c_1-c_2) t+c_2) \end{align*}

✓ Sympy. Time used: 0.128 (sec). Leaf size: 78

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
ode=[Eq(-10*x__1(t) + 4*x__2(t) + Derivative(x__1(t), t),0),Eq(-4*x__1(t) - 2*x__2(t) + Derivative(x__2(t), t) - exp(6*t)/t,0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
 

\[ \left [ x^{1}{\left (t \right )} = t \left (4 C_{1} + 4\right ) e^{6 t} - 4 t e^{6 t} \log {\left (t \right )} + \left (C_{1} + 4 C_{2}\right ) e^{6 t}, \ x^{2}{\left (t \right )} = 4 C_{2} e^{6 t} + t \left (4 C_{1} + 4\right ) e^{6 t} - 4 t e^{6 t} \log {\left (t \right )} + e^{6 t} \log {\left (t \right )}\right ] \]