63.4.25 problem 13

Internal problem ID [13068]
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
Date solved : Tuesday, January 28, 2025 at 04:51:08 AM
CAS classification : [_separable]

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

With initial conditions

\begin{align*} x \left (1\right )&=1 \end{align*}

✓ Solution by Maple

Time used: 0.102 (sec). Leaf size: 120

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 )^{{2}/{3}}+4}{2 \left (-4-4 t^{3}+4 \sqrt {t^{6}+2 t^{3}-3}\right )^{{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 )^{{2}/{3}}-4 i \sqrt {3}+4}{4 \left (-4-4 t^{3}+4 \sqrt {t^{6}+2 t^{3}-3}\right )^{{1}/{3}}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 2.603 (sec). Leaf size: 188

DSolve[{D[x[t],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*}