76.1.9 problem 9
Internal
problem
ID
[17237]
Book
:
Differential
equations.
An
introduction
to
modern
methods
and
applications.
James
Brannan,
William
E.
Boyce.
Third
edition.
Wiley
2015
Section
:
Chapter
2.
First
order
differential
equations.
Section
2.1
(Separable
equations).
Problems
at
page
44
Problem
number
:
9
Date
solved
:
Monday, March 31, 2025 at 03:45:34 PM
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 \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 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: 2.27 (sec). Leaf size: 307
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 {-2+\sqrt [3]{2} \left (x^3+\sqrt {x^6+6 c_1 x^3+4+9 c_1{}^2}+3 c_1\right ){}^{2/3}}{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 ) \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 {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}}-\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: 23.381 (sec). Leaf size: 337
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-x**2/(y(x)**2 + 1) + Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = \sqrt [3]{2} \left (\frac {\sqrt [3]{2} \sqrt [3]{- 3 C_{1} - x^{3} + \sqrt {9 C_{1}^{2} + 6 C_{1} x^{3} + x^{6} + 4}}}{4} + \frac {\sqrt [3]{2} \sqrt {3} i \sqrt [3]{- 3 C_{1} - x^{3} + \sqrt {9 C_{1}^{2} + 6 C_{1} x^{3} + x^{6} + 4}}}{4} + \frac {2}{\left (-1 - \sqrt {3} i\right ) \sqrt [3]{- 3 C_{1} - x^{3} + \sqrt {9 C_{1}^{2} + 6 C_{1} x^{3} + x^{6} + 4}}}\right ), \ y{\left (x \right )} = \sqrt [3]{2} \left (\frac {\sqrt [3]{2} \sqrt [3]{- 3 C_{1} - x^{3} + \sqrt {9 C_{1}^{2} + 6 C_{1} x^{3} + x^{6} + 4}}}{4} - \frac {\sqrt [3]{2} \sqrt {3} i \sqrt [3]{- 3 C_{1} - x^{3} + \sqrt {9 C_{1}^{2} + 6 C_{1} x^{3} + x^{6} + 4}}}{4} + \frac {2}{\left (-1 + \sqrt {3} i\right ) \sqrt [3]{- 3 C_{1} - x^{3} + \sqrt {9 C_{1}^{2} + 6 C_{1} x^{3} + x^{6} + 4}}}\right ), \ y{\left (x \right )} = \sqrt [3]{2} \left (- \frac {\sqrt [3]{2} \sqrt [3]{- 3 C_{1} - x^{3} + \sqrt {9 C_{1}^{2} + 6 C_{1} x^{3} + x^{6} + 4}}}{2} + \frac {1}{\sqrt [3]{- 3 C_{1} - x^{3} + \sqrt {9 C_{1}^{2} + 6 C_{1} x^{3} + x^{6} + 4}}}\right )\right ]
\]