32.4.21 problem Recognizable Exact Differential equations. Integrating factors. Exercise 10.13, page 90
Internal
problem
ID
[5832]
Book
:
Ordinary
Differential
Equations,
By
Tenenbaum
and
Pollard.
Dover,
NY
1963
Section
:
Chapter
2.
Special
types
of
differential
equations
of
the
first
kind.
Lesson
10
Problem
number
:
Recognizable
Exact
Differential
equations.
Integrating
factors.
Exercise
10.13,
page
90
Date
solved
:
Monday, January 27, 2025 at 01:20:12 PM
CAS
classification
:
[_rational, [_Abel, `2nd type`, `class B`]]
\begin{align*} y \left (y+2 x +1\right )-x \left (x +2 y-1\right ) y^{\prime }&=0 \end{align*}
✓ Solution by Maple
Time used: 0.002 (sec). Leaf size: 385
dsolve((y(x)*(y(x)+2*x+1))-(x*(2*y(x)+x-1))*diff(y(x),x)=0,y(x), singsol=all)
\begin{align*}
y \left (x \right ) &= \frac {3 \,5^{{1}/{3}} {\left (x \left (\sqrt {5}\, \sqrt {\frac {80 c_{1} x^{2}-160 c_{1} x +80 c_{1} -x}{c_{1}}}+20 x -20\right ) c_{1}^{2}\right )}^{{1}/{3}}}{40 c_{1}}+\frac {3 x 5^{{2}/{3}}}{40 {\left (x \left (\sqrt {5}\, \sqrt {\frac {80 c_{1} x^{2}-160 c_{1} x +80 c_{1} -x}{c_{1}}}+20 x -20\right ) c_{1}^{2}\right )}^{{1}/{3}}}+x -1 \\
y \left (x \right ) &= \frac {\frac {3 \left (-1-i \sqrt {3}\right ) 5^{{1}/{3}} {\left (x \left (\sqrt {5}\, \sqrt {\frac {80 \left (x -1\right )^{2} c_{1} -x}{c_{1}}}+20 x -20\right ) c_{1}^{2}\right )}^{{2}/{3}}}{80}+\frac {3 c_{1} \left (\frac {80 \left (x -1\right ) {\left (x \left (\sqrt {5}\, \sqrt {\frac {80 \left (x -1\right )^{2} c_{1} -x}{c_{1}}}+20 x -20\right ) c_{1}^{2}\right )}^{{1}/{3}}}{3}+x 5^{{2}/{3}} \left (i \sqrt {3}-1\right )\right )}{80}}{c_{1} {\left (x \left (\sqrt {5}\, \sqrt {\frac {80 \left (x -1\right )^{2} c_{1} -x}{c_{1}}}+20 x -20\right ) c_{1}^{2}\right )}^{{1}/{3}}} \\
y \left (x \right ) &= \frac {\frac {3 \left (i \sqrt {3}-1\right ) 5^{{1}/{3}} {\left (x \left (\sqrt {5}\, \sqrt {\frac {80 \left (x -1\right )^{2} c_{1} -x}{c_{1}}}+20 x -20\right ) c_{1}^{2}\right )}^{{2}/{3}}}{80}+\frac {3 \left (-\frac {80 \left (1-x \right ) {\left (x \left (\sqrt {5}\, \sqrt {\frac {80 \left (x -1\right )^{2} c_{1} -x}{c_{1}}}+20 x -20\right ) c_{1}^{2}\right )}^{{1}/{3}}}{3}+\left (-1-i \sqrt {3}\right ) x 5^{{2}/{3}}\right ) c_{1}}{80}}{c_{1} {\left (x \left (\sqrt {5}\, \sqrt {\frac {80 \left (x -1\right )^{2} c_{1} -x}{c_{1}}}+20 x -20\right ) c_{1}^{2}\right )}^{{1}/{3}}} \\
\end{align*}
✓ Solution by Mathematica
Time used: 37.819 (sec). Leaf size: 463
DSolve[(y[x]*(y[x]+2*x+1))-(x*(2*y[x]+x-1))*D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to -\frac {\sqrt [3]{2} x}{\sqrt [3]{-27 c_1{}^2 x^2+\sqrt {108 c_1{}^3 x^3+\left (27 c_1{}^2 x-27 c_1{}^2 x^2\right ){}^2}+27 c_1{}^2 x}}+\frac {\sqrt [3]{-27 c_1{}^2 x^2+\sqrt {108 c_1{}^3 x^3+\left (27 c_1{}^2 x-27 c_1{}^2 x^2\right ){}^2}+27 c_1{}^2 x}}{3 \sqrt [3]{2} c_1}+x-1 \\
y(x)\to \frac {\left (1+i \sqrt {3}\right ) x}{2^{2/3} \sqrt [3]{-27 c_1{}^2 x^2+\sqrt {108 c_1{}^3 x^3+\left (27 c_1{}^2 x-27 c_1{}^2 x^2\right ){}^2}+27 c_1{}^2 x}}-\frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{-27 c_1{}^2 x^2+\sqrt {108 c_1{}^3 x^3+\left (27 c_1{}^2 x-27 c_1{}^2 x^2\right ){}^2}+27 c_1{}^2 x}}{6 \sqrt [3]{2} c_1}+x-1 \\
y(x)\to \frac {\left (1-i \sqrt {3}\right ) x}{2^{2/3} \sqrt [3]{-27 c_1{}^2 x^2+\sqrt {108 c_1{}^3 x^3+\left (27 c_1{}^2 x-27 c_1{}^2 x^2\right ){}^2}+27 c_1{}^2 x}}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{-27 c_1{}^2 x^2+\sqrt {108 c_1{}^3 x^3+\left (27 c_1{}^2 x-27 c_1{}^2 x^2\right ){}^2}+27 c_1{}^2 x}}{6 \sqrt [3]{2} c_1}+x-1 \\
y(x)\to \text {Indeterminate} \\
y(x)\to x-1 \\
\end{align*}