Internal problem ID [10374]
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 1.3.1 Separable equations. Exercises page
26
Problem number: 13.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_separable]
\[ \boxed {x^{\prime }-\frac {t^{2}}{1-x^{2}}=0} \] With initial conditions \begin {align*} [x \left (1\right ) = 1] \end {align*}
✓ Solution by Maple
Time used: 0.094 (sec). Leaf size: 122
dsolve([diff(x(t),t)=t^2/(1-x(t)^2),x(1) = 1],x(t), singsol=all)
\begin{align*} x \left (t \right ) = \frac {\left (-4-4 t^{3}+4 \sqrt {t^{6}+2 t^{3}-3}\right )^{\frac {2}{3}}+4}{2 \left (-4-4 t^{3}+4 \sqrt {t^{6}+2 t^{3}-3}\right )^{\frac {1}{3}}} \\ x \left (t \right ) = -\frac {\left (1+i \sqrt {3}\right ) \left (-4-4 t^{3}+4 \sqrt {t^{6}+2 t^{3}-3}\right )^{\frac {2}{3}}-4 i \sqrt {3}+4}{4 \left (-4-4 t^{3}+4 \sqrt {t^{6}+2 t^{3}-3}\right )^{\frac {1}{3}}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 2.555 (sec). Leaf size: 188
DSolve[{x'[t]==t^2/(1-x[t]^2),{x[1]==1}},x[t],t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to \frac {\sqrt [3]{-t^3+\sqrt {t^6+2 t^3-3}-1}}{\sqrt [3]{2}}+\frac {\sqrt [3]{2}}{\sqrt [3]{-t^3+\sqrt {t^6+2 t^3-3}-1}} \\ x(t)\to \frac {-i \sqrt [3]{2} \sqrt {3} \left (-t^3+\sqrt {t^6+2 t^3-3}-1\right )^{2/3}-\sqrt [3]{2} \left (-t^3+\sqrt {t^6+2 t^3-3}-1\right )^{2/3}+2 i \sqrt {3}-2}{2\ 2^{2/3} \sqrt [3]{-t^3+\sqrt {t^6+2 t^3-3}-1}} \\ \end{align*}