63.5.28 problem 15(c)

Internal problem ID [13102]
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 : Tuesday, January 28, 2025 at 04:52:23 AM
CAS classification : [_separable]

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

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 59

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

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

✓ Solution by Mathematica

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

DSolve[D[x[t],t]==-1/t*x[t]+1/(t*x[t]^2),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*}