66.1.17 problem Problem 17
Internal
problem
ID
[13864]
Book
:
Differential
equations
and
the
calculus
of
variations
by
L.
ElSGOLTS.
MIR
PUBLISHERS,
MOSCOW,
Third
printing
1977.
Section
:
Chapter
1,
First-Order
Differential
Equations.
Problems
page
88
Problem
number
:
Problem
17
Date
solved
:
Tuesday, January 28, 2025 at 06:06:02 AM
CAS
classification
:
[[_homogeneous, `class G`], _rational]
\begin{align*} y^{\prime }&=\frac {y}{x +y^{3}} \end{align*}
✓ Solution by Maple
Time used: 0.003 (sec). Leaf size: 220
dsolve(diff(y(x),x)=y(x)/(x+y(x)^3),y(x), singsol=all)
\begin{align*}
y &= \frac {\left (27 x +3 \sqrt {24 c_{1}^{3}+81 x^{2}}\right )^{{2}/{3}}-6 c_{1}}{3 \left (27 x +3 \sqrt {24 c_{1}^{3}+81 x^{2}}\right )^{{1}/{3}}} \\
y &= -\frac {i \sqrt {3}\, \left (27 x +3 \sqrt {24 c_{1}^{3}+81 x^{2}}\right )^{{2}/{3}}+6 i \sqrt {3}\, c_{1} +\left (27 x +3 \sqrt {24 c_{1}^{3}+81 x^{2}}\right )^{{2}/{3}}-6 c_{1}}{6 \left (27 x +3 \sqrt {24 c_{1}^{3}+81 x^{2}}\right )^{{1}/{3}}} \\
y &= \frac {i \sqrt {3}\, \left (27 x +3 \sqrt {24 c_{1}^{3}+81 x^{2}}\right )^{{2}/{3}}+6 i \sqrt {3}\, c_{1} -\left (27 x +3 \sqrt {24 c_{1}^{3}+81 x^{2}}\right )^{{2}/{3}}+6 c_{1}}{6 \left (27 x +3 \sqrt {24 c_{1}^{3}+81 x^{2}}\right )^{{1}/{3}}} \\
\end{align*}
✓ Solution by Mathematica
Time used: 1.855 (sec). Leaf size: 263
DSolve[D[y[x],x]==y[x]/(x+y[x]^3),y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \frac {2\ 3^{2/3} c_1-\sqrt [3]{3} \left (-9 x+\sqrt {81 x^2+24 c_1{}^3}\right ){}^{2/3}}{3 \sqrt [3]{-9 x+\sqrt {81 x^2+24 c_1{}^3}}} \\
y(x)\to \frac {\sqrt [3]{3} \left (1-i \sqrt {3}\right ) \left (-9 x+\sqrt {81 x^2+24 c_1{}^3}\right ){}^{2/3}-2 \sqrt [6]{3} \left (\sqrt {3}+3 i\right ) c_1}{6 \sqrt [3]{-9 x+\sqrt {81 x^2+24 c_1{}^3}}} \\
y(x)\to \frac {\sqrt [3]{3} \left (1+i \sqrt {3}\right ) \left (-9 x+\sqrt {81 x^2+24 c_1{}^3}\right ){}^{2/3}-2 \sqrt [6]{3} \left (\sqrt {3}-3 i\right ) c_1}{6 \sqrt [3]{-9 x+\sqrt {81 x^2+24 c_1{}^3}}} \\
y(x)\to 0 \\
\end{align*}