14.15.7 problem 7

Internal problem ID [3833]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 9, First order linear systems. Section 9.3, page 598
Problem number : 7
Date solved : Sunday, October 12, 2025 at 01:17:49 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.123 (sec). Leaf size: 29

ode:=[diff(x__1(t),t) = 1/t*x__1(t)+t*x__2(t), diff(x__2(t),t) = -1/t*x__1(t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= -\left (c_1 \cos \left (t \right )-c_2 \sin \left (t \right )\right ) t \\ x_{2} \left (t \right ) &= c_1 \sin \left (t \right )+c_2 \cos \left (t \right ) \\ \end{align*}

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 33

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

\begin{align*} \text {x1}(t)&\to c_1 t \sin (t)-c_2 t \cos (t)\\ \text {x2}(t)&\to c_1 \cos (t)+c_2 \sin (t) \end{align*}

✓ Sympy. Time used: 0.248 (sec). Leaf size: 27

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

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