63.5.28 problem 15(c)

Internal problem ID [13031]
Book : A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY. 2015.
Section : Chapter 1, First order differential equations. Section 1.4.1. Integrating factors. Exercises page 41
Problem number : 15(c)
Date solved : Monday, March 31, 2025 at 07:31:34 AM
CAS classification : [_separable]

\begin{align*} x^{\prime }&=-\frac {x}{t}+\frac {1}{t x^{2}} \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 59

ode:=diff(x(t),t) = -x(t)/t+1/t/x(t)^2; 
dsolve(ode,x(t), singsol=all);
 

\begin{align*} x &= \frac {\left (t^{3}+c_1 \right )^{{1}/{3}}}{t} \\ x &= -\frac {\left (t^{3}+c_1 \right )^{{1}/{3}} \left (1+i \sqrt {3}\right )}{2 t} \\ x &= \frac {\left (t^{3}+c_1 \right )^{{1}/{3}} \left (i \sqrt {3}-1\right )}{2 t} \\ \end{align*}

✓ Mathematica. Time used: 0.293 (sec). Leaf size: 159

ode=D[x[t],t]==-1/t*x[t]+1/(t*x[t]^2); 
ic={}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)\to \frac {\sqrt [3]{t^3+e^{3 c_1}}}{t} \\ x(t)\to -\frac {\sqrt [3]{-1} \sqrt [3]{t^3+e^{3 c_1}}}{t} \\ x(t)\to \frac {(-1)^{2/3} \sqrt [3]{t^3+e^{3 c_1}}}{t} \\ x(t)\to 1 \\ x(t)\to -\sqrt [3]{-1} \\ x(t)\to (-1)^{2/3} \\ x(t)\to \frac {\sqrt [3]{t^3}}{t} \\ x(t)\to -\frac {\sqrt [3]{-1} \sqrt [3]{t^3}}{t} \\ x(t)\to \frac {(-1)^{2/3} \sqrt [3]{t^3}}{t} \\ \end{align*}

✓ Sympy. Time used: 1.472 (sec). Leaf size: 58

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(Derivative(x(t), t) + x(t)/t - 1/(t*x(t)**2),0) 
ics = {} 
dsolve(ode,func=x(t),ics=ics)
 

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