76.1.9 problem 9
Internal
problem
ID
[17308]
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
:
Tuesday, January 28, 2025 at 09:59:21 AM
CAS
classification
:
[_separable]
\begin{align*} y^{\prime }&=\frac {x^{2}}{1+y^{2}} \end{align*}
✓ Solution by Maple
Time used: 0.002 (sec). Leaf size: 264
dsolve(diff(y(x),x)=x^2/(1+y(x)^2),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*}
✓ Solution by Mathematica
Time used: 2.270 (sec). Leaf size: 307
DSolve[D[y[x],x]==x^2/(1+y[x]^2),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*}