Internal problem ID [11436]
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).
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_separable]
\[ \boxed {x^{\prime }+\frac {x}{t}-\frac {1}{t x^{2}}=0} \]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 75
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 )^{\frac {1}{3}}}{t} x \left (t \right ) = \frac {-\frac {\left (t^{3}+c_{1} \right )^{\frac {1}{3}}}{2}-\frac {i \sqrt {3}\, \left (t^{3}+c_{1} \right )^{\frac {1}{3}}}{2}}{t} x \left (t \right ) = \frac {-\frac {\left (t^{3}+c_{1} \right )^{\frac {1}{3}}}{2}+\frac {i \sqrt {3}\, \left (t^{3}+c_{1} \right )^{\frac {1}{3}}}{2}}{t} \end{align*}
✓ Solution by Mathematica
Time used: 0.456 (sec). Leaf size: 159
DSolve[x'[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*}