32.6.43 problem Exercise 12.43, page 103
Internal
problem
ID
[5908]
Book
:
Ordinary
Differential
Equations,
By
Tenenbaum
and
Pollard.
Dover,
NY
1963
Section
:
Chapter
2.
Special
types
of
differential
equations
of
the
first
kind.
Lesson
12,
Miscellaneous
Methods
Problem
number
:
Exercise
12.43,
page
103
Date
solved
:
Monday, January 27, 2025 at 01:26:43 PM
CAS
classification
:
[[_homogeneous, `class A`], _rational, _dAlembert]
\begin{align*} \left (2 x y^{3}-x^{4}\right ) y^{\prime }+2 x^{3} y-y^{4}&=0 \end{align*}
✓ Solution by Maple
Time used: 0.059 (sec). Leaf size: 313
dsolve((2*x*y(x)^3-x^4)*diff(y(x),x)+2*x^3*y(x)-y(x)^4=0,y(x), singsol=all)
\begin{align*}
y \left (x \right ) &= \frac {12^{{1}/{3}} \left (x 12^{{1}/{3}} c_{1} +{\left (x \left (-9 c_{1} x^{2}+\sqrt {3}\, \sqrt {\frac {27 c_{1}^{3} x^{4}-4 x}{c_{1}}}\right ) c_{1}^{2}\right )}^{{2}/{3}}\right )}{6 c_{1} {\left (x \left (-9 c_{1} x^{2}+\sqrt {3}\, \sqrt {\frac {27 c_{1}^{3} x^{4}-4 x}{c_{1}}}\right ) c_{1}^{2}\right )}^{{1}/{3}}} \\
y \left (x \right ) &= \frac {2^{{2}/{3}} 3^{{1}/{3}} \left (\left (-1-i \sqrt {3}\right ) {\left (x \left (-9 c_{1} x^{2}+\sqrt {3}\, \sqrt {\frac {27 c_{1}^{3} x^{4}-4 x}{c_{1}}}\right ) c_{1}^{2}\right )}^{{2}/{3}}+2^{{2}/{3}} x \left (i 3^{{5}/{6}}-3^{{1}/{3}}\right ) c_{1} \right )}{12 {\left (x \left (-9 c_{1} x^{2}+\sqrt {3}\, \sqrt {\frac {27 c_{1}^{3} x^{4}-4 x}{c_{1}}}\right ) c_{1}^{2}\right )}^{{1}/{3}} c_{1}} \\
y \left (x \right ) &= -\frac {2^{{2}/{3}} \left (\left (1-i \sqrt {3}\right ) {\left (x \left (-9 c_{1} x^{2}+\sqrt {3}\, \sqrt {\frac {27 c_{1}^{3} x^{4}-4 x}{c_{1}}}\right ) c_{1}^{2}\right )}^{{2}/{3}}+2^{{2}/{3}} x \left (i 3^{{5}/{6}}+3^{{1}/{3}}\right ) c_{1} \right ) 3^{{1}/{3}}}{12 {\left (x \left (-9 c_{1} x^{2}+\sqrt {3}\, \sqrt {\frac {27 c_{1}^{3} x^{4}-4 x}{c_{1}}}\right ) c_{1}^{2}\right )}^{{1}/{3}} c_{1}} \\
\end{align*}
✓ Solution by Mathematica
Time used: 60.235 (sec). Leaf size: 331
DSolve[(2*x*y[x]^3-x^4)*D[y[x],x]+2*x^3*y[x]-y[x]^4==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \frac {\sqrt [3]{2} \left (-9 x^3+\sqrt {81 x^6-12 e^{3 c_1} x^3}\right ){}^{2/3}+2 \sqrt [3]{3} e^{c_1} x}{6^{2/3} \sqrt [3]{-9 x^3+\sqrt {81 x^6-12 e^{3 c_1} x^3}}} \\
y(x)\to \frac {i \sqrt [3]{2} \sqrt [6]{3} \left (\sqrt {3}+i\right ) \left (-9 x^3+\sqrt {81 x^6-12 e^{3 c_1} x^3}\right ){}^{2/3}-2 \left (\sqrt {3}+3 i\right ) e^{c_1} x}{2\ 2^{2/3} 3^{5/6} \sqrt [3]{-9 x^3+\sqrt {81 x^6-12 e^{3 c_1} x^3}}} \\
y(x)\to \frac {\sqrt [3]{2} \sqrt [6]{3} \left (-1-i \sqrt {3}\right ) \left (-9 x^3+\sqrt {81 x^6-12 e^{3 c_1} x^3}\right ){}^{2/3}-2 \left (\sqrt {3}-3 i\right ) e^{c_1} x}{2\ 2^{2/3} 3^{5/6} \sqrt [3]{-9 x^3+\sqrt {81 x^6-12 e^{3 c_1} x^3}}} \\
\end{align*}