57.4.25 problem 13

Internal problem ID [14349]
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 : Thursday, October 02, 2025 at 09:32:08 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*}

✓ Maple. Time used: 0.138 (sec). Leaf size: 120

ode:=diff(x(t),t) = t^2/(1-x(t)^2); 
ic:=[x(1) = 1]; 
dsolve([ode,op(ic)],x(t), singsol=all);
 

\begin{align*} x &= \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 &= -\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*}

✓ Mathematica. Time used: 1.93 (sec). Leaf size: 188

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

✓ Sympy. Time used: 14.973 (sec). Leaf size: 209

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

\[ \left [ x{\left (t \right )} = \sqrt [3]{2} \left (\frac {\sqrt [3]{2} \sqrt [3]{t^{3} + \sqrt {t^{6} + 2 t^{3} - 3} + 1}}{4} + \frac {\sqrt [3]{2} \sqrt {3} i \sqrt [3]{t^{3} + \sqrt {t^{6} + 2 t^{3} - 3} + 1}}{4} - \frac {2}{\left (-1 - \sqrt {3} i\right ) \sqrt [3]{t^{3} + \sqrt {t^{6} + 2 t^{3} - 3} + 1}}\right ), \ x{\left (t \right )} = \sqrt [3]{2} \left (\frac {\sqrt [3]{2} \sqrt [3]{t^{3} + \sqrt {t^{6} + 2 t^{3} - 3} + 1}}{4} - \frac {\sqrt [3]{2} \sqrt {3} i \sqrt [3]{t^{3} + \sqrt {t^{6} + 2 t^{3} - 3} + 1}}{4} - \frac {2}{\left (-1 + \sqrt {3} i\right ) \sqrt [3]{t^{3} + \sqrt {t^{6} + 2 t^{3} - 3} + 1}}\right )\right ] \]