63.4.25 problem 13
Internal
problem
ID
[12989]
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
:
Wednesday, March 05, 2025 at 08:56:17 PM
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: 0.102 (sec). Leaf size: 120
ode:=diff(x(t),t) = t^2/(1-x(t)^2);
ic:=x(1) = 1;
dsolve([ode,ic],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*}
✓ Mathematica. Time used: 2.603 (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: 21.630 (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)
\[
\left [ x{\left (t \right )} = \sqrt [3]{2} \left (\frac {\sqrt [3]{2} \sqrt [3]{t^{3} + \sqrt {t^{6} + 2 t^{3} - 3} + 1}}{4} + \frac {\sqrt [3]{2} \sqrt {3} i \sqrt [3]{t^{3} + \sqrt {t^{6} + 2 t^{3} - 3} + 1}}{4} - \frac {2}{\left (-1 - \sqrt {3} i\right ) \sqrt [3]{t^{3} + \sqrt {t^{6} + 2 t^{3} - 3} + 1}}\right ), \ x{\left (t \right )} = \sqrt [3]{2} \left (\frac {\sqrt [3]{2} \sqrt [3]{t^{3} + \sqrt {t^{6} + 2 t^{3} - 3} + 1}}{4} - \frac {\sqrt [3]{2} \sqrt {3} i \sqrt [3]{t^{3} + \sqrt {t^{6} + 2 t^{3} - 3} + 1}}{4} - \frac {2}{\left (-1 + \sqrt {3} i\right ) \sqrt [3]{t^{3} + \sqrt {t^{6} + 2 t^{3} - 3} + 1}}\right )\right ]
\]