20.15.6 problem 6

Internal problem ID [3832]
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 : 6
Date solved : Friday, March 14, 2025 at 01:27:28 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.020 (sec). Leaf size: 14

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

\begin{align*} x_{1} \left (t \right ) &= c_{2} t \\ x_{2} \left (t \right ) &= {\mathrm e}^{t} c_{1} \\ \end{align*}

✓ Mathematica. Time used: 0.041 (sec). Leaf size: 53

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

\begin{align*} \text {x1}(t)\to c_1 t \\ \text {x2}(t)\to c_2 e^t \\ \text {x1}(t)\to c_1 t \\ \text {x2}(t)\to 0 \\ \text {x1}(t)\to 0 \\ \text {x2}(t)\to c_2 e^t \\ \text {x1}(t)\to 0 \\ \text {x2}(t)\to 0 \\ \end{align*}

✓ Sympy. Time used: 0.148 (sec). Leaf size: 12

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

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