48.1.2 problem Example 3.2
Internal
problem
ID
[7503]
Book
:
THEORY
OF
DIFFERENTIAL
EQUATIONS
IN
ENGINEERING
AND
MECHANICS.
K.T.
CHAU,
CRC
Press.
Boca
Raton,
FL.
2018
Section
:
Chapter
3.
Ordinary
Differential
Equations.
Section
3.2
FIRST
ORDER
ODE.
Page
114
Problem
number
:
Example
3.2
Date
solved
:
Wednesday, March 05, 2025 at 04:41:11 AM
CAS
classification
:
[_separable]
\begin{align*} y^{\prime }&=\frac {x^{2}}{1-y^{2}} \end{align*}
✓ Maple. Time used: 0.003 (sec). Leaf size: 264
ode:=diff(y(x),x) = x^2/(1-y(x)^2);
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {\left (-4 x^{3}-12 c_{1} +4 \sqrt {x^{6}+6 c_{1} x^{3}+9 c_{1}^{2}-4}\right )^{{2}/{3}}+4}{2 \left (-4 x^{3}-12 c_{1} +4 \sqrt {x^{6}+6 c_{1} x^{3}+9 c_{1}^{2}-4}\right )^{{1}/{3}}} \\
y &= -\frac {\left (1+i \sqrt {3}\right ) \left (-4 x^{3}-12 c_{1} +4 \sqrt {x^{6}+6 c_{1} x^{3}+9 c_{1}^{2}-4}\right )^{{2}/{3}}-4 i \sqrt {3}+4}{4 \left (-4 x^{3}-12 c_{1} +4 \sqrt {x^{6}+6 c_{1} x^{3}+9 c_{1}^{2}-4}\right )^{{1}/{3}}} \\
y &= \frac {i \left (-4 x^{3}-12 c_{1} +4 \sqrt {x^{6}+6 c_{1} x^{3}+9 c_{1}^{2}-4}\right )^{{2}/{3}} \sqrt {3}-4 i \sqrt {3}-\left (-4 x^{3}-12 c_{1} +4 \sqrt {x^{6}+6 c_{1} x^{3}+9 c_{1}^{2}-4}\right )^{{2}/{3}}-4}{4 \left (-4 x^{3}-12 c_{1} +4 \sqrt {x^{6}+6 c_{1} x^{3}+9 c_{1}^{2}-4}\right )^{{1}/{3}}} \\
\end{align*}
✓ Mathematica. Time used: 2.581 (sec). Leaf size: 320
ode=D[y[x],x]==x^2/(1-y[x]^2);
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \frac {\sqrt [3]{-x^3+\sqrt {x^6-6 c_1 x^3-4+9 c_1{}^2}+3 c_1}}{\sqrt [3]{2}}+\frac {\sqrt [3]{2}}{\sqrt [3]{-x^3+\sqrt {x^6-6 c_1 x^3-4+9 c_1{}^2}+3 c_1}} \\
y(x)\to \frac {i \left (\sqrt {3}+i\right ) \sqrt [3]{-x^3+\sqrt {x^6-6 c_1 x^3-4+9 c_1{}^2}+3 c_1}}{2 \sqrt [3]{2}}-\frac {1+i \sqrt {3}}{2^{2/3} \sqrt [3]{-x^3+\sqrt {x^6-6 c_1 x^3-4+9 c_1{}^2}+3 c_1}} \\
y(x)\to \frac {i \left (\sqrt {3}+i\right )}{2^{2/3} \sqrt [3]{-x^3+\sqrt {x^6-6 c_1 x^3-4+9 c_1{}^2}+3 c_1}}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{-x^3+\sqrt {x^6-6 c_1 x^3-4+9 c_1{}^2}+3 c_1}}{2 \sqrt [3]{2}} \\
\end{align*}
✓ Sympy. Time used: 21.832 (sec). Leaf size: 338
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-x**2/(1 - y(x)**2) + Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = \sqrt [3]{2} \left (\frac {\sqrt [3]{2} \sqrt [3]{- 3 C_{1} + x^{3} + \sqrt {9 C_{1}^{2} - 6 C_{1} x^{3} + x^{6} - 4}}}{4} + \frac {\sqrt [3]{2} \sqrt {3} i \sqrt [3]{- 3 C_{1} + x^{3} + \sqrt {9 C_{1}^{2} - 6 C_{1} x^{3} + x^{6} - 4}}}{4} - \frac {2}{\left (-1 - \sqrt {3} i\right ) \sqrt [3]{- 3 C_{1} + x^{3} + \sqrt {9 C_{1}^{2} - 6 C_{1} x^{3} + x^{6} - 4}}}\right ), \ y{\left (x \right )} = \sqrt [3]{2} \left (\frac {\sqrt [3]{2} \sqrt [3]{- 3 C_{1} + x^{3} + \sqrt {9 C_{1}^{2} - 6 C_{1} x^{3} + x^{6} - 4}}}{4} - \frac {\sqrt [3]{2} \sqrt {3} i \sqrt [3]{- 3 C_{1} + x^{3} + \sqrt {9 C_{1}^{2} - 6 C_{1} x^{3} + x^{6} - 4}}}{4} - \frac {2}{\left (-1 + \sqrt {3} i\right ) \sqrt [3]{- 3 C_{1} + x^{3} + \sqrt {9 C_{1}^{2} - 6 C_{1} x^{3} + x^{6} - 4}}}\right ), \ y{\left (x \right )} = \sqrt [3]{2} \left (- \frac {\sqrt [3]{2} \sqrt [3]{- 3 C_{1} + x^{3} + \sqrt {9 C_{1}^{2} - 6 C_{1} x^{3} + x^{6} - 4}}}{2} - \frac {1}{\sqrt [3]{- 3 C_{1} + x^{3} + \sqrt {9 C_{1}^{2} - 6 C_{1} x^{3} + x^{6} - 4}}}\right )\right ]
\]