32.4.9 problem Recognizable Exact Differential equations. Integrating factors. Exercise 10.1, page 90
Internal
problem
ID
[5820]
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
10.1,
page
90
Date
solved
:
Monday, January 27, 2025 at 01:20:02 PM
CAS
classification
:
[[_homogeneous, `class A`], _exact, _rational, _dAlembert]
\begin{align*} 2 y x +x^{2}+\left (x^{2}+y^{2}\right ) y^{\prime }&=0 \end{align*}
✓ Solution by Maple
Time used: 0.065 (sec). Leaf size: 317
dsolve((2*x*y(x)+x^2)+(x^2+y(x)^2)*diff(y(x),x)=0,y(x), singsol=all)
\begin{align*}
y \left (x \right ) &= -\frac {2 \left (c_{1} x^{2}-\frac {\left (4-4 x^{3} c_{1}^{{3}/{2}}+4 \sqrt {5 x^{6} c_{1}^{3}-2 x^{3} c_{1}^{{3}/{2}}+1}\right )^{{2}/{3}}}{4}\right )}{\sqrt {c_{1}}\, \left (4-4 x^{3} c_{1}^{{3}/{2}}+4 \sqrt {5 x^{6} c_{1}^{3}-2 x^{3} c_{1}^{{3}/{2}}+1}\right )^{{1}/{3}}} \\
y \left (x \right ) &= -\frac {\left (1+i \sqrt {3}\right ) \left (4-4 x^{3} c_{1}^{{3}/{2}}+4 \sqrt {5 x^{6} c_{1}^{3}-2 x^{3} c_{1}^{{3}/{2}}+1}\right )^{{1}/{3}}}{4 \sqrt {c_{1}}}-\frac {x^{2} \sqrt {c_{1}}\, \left (i \sqrt {3}-1\right )}{\left (4-4 x^{3} c_{1}^{{3}/{2}}+4 \sqrt {5 x^{6} c_{1}^{3}-2 x^{3} c_{1}^{{3}/{2}}+1}\right )^{{1}/{3}}} \\
y \left (x \right ) &= \frac {4 i \sqrt {3}\, c_{1} x^{2}+i \sqrt {3}\, \left (4-4 x^{3} c_{1}^{{3}/{2}}+4 \sqrt {5 x^{6} c_{1}^{3}-2 x^{3} c_{1}^{{3}/{2}}+1}\right )^{{2}/{3}}+4 c_{1} x^{2}-\left (4-4 x^{3} c_{1}^{{3}/{2}}+4 \sqrt {5 x^{6} c_{1}^{3}-2 x^{3} c_{1}^{{3}/{2}}+1}\right )^{{2}/{3}}}{4 \left (4-4 x^{3} c_{1}^{{3}/{2}}+4 \sqrt {5 x^{6} c_{1}^{3}-2 x^{3} c_{1}^{{3}/{2}}+1}\right )^{{1}/{3}} \sqrt {c_{1}}} \\
\end{align*}
✓ Solution by Mathematica
Time used: 36.719 (sec). Leaf size: 597
DSolve[(2*x*y[x]+x^2)+(x^2+y[x]^2)*D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*}
y(x)\to \frac {\sqrt [3]{-x^3+\sqrt {5 x^6-2 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}}{\sqrt [3]{2}}-\frac {\sqrt [3]{2} x^2}{\sqrt [3]{-x^3+\sqrt {5 x^6-2 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}} \\
y(x)\to \frac {\sqrt [3]{2} \left (2+2 i \sqrt {3}\right ) x^2+i 2^{2/3} \left (\sqrt {3}+i\right ) \left (-x^3+\sqrt {5 x^6-2 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}\right ){}^{2/3}}{4 \sqrt [3]{-x^3+\sqrt {5 x^6-2 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}} \\
y(x)\to \frac {\left (1-i \sqrt {3}\right ) x^2}{2^{2/3} \sqrt [3]{-x^3+\sqrt {5 x^6-2 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{-x^3+\sqrt {5 x^6-2 e^{3 c_1} x^3+e^{6 c_1}}+e^{3 c_1}}}{2 \sqrt [3]{2}} \\
y(x)\to \frac {2 \sqrt [3]{-2} x^2+(-2)^{2/3} \left (\sqrt {5} \sqrt {x^6}-x^3\right )^{2/3}}{2 \sqrt [3]{\sqrt {5} \sqrt {x^6}-x^3}} \\
y(x)\to \frac {\left (2 \sqrt {5} \sqrt {x^6}-2 x^3\right )^{2/3}-2 \sqrt [3]{2} x^2}{2 \sqrt [3]{\sqrt {5} \sqrt {x^6}-x^3}} \\
y(x)\to \frac {\sqrt [3]{2} \left (2-2 i \sqrt {3}\right ) x^2+\left (-1-i \sqrt {3}\right ) \left (2 \sqrt {5} \sqrt {x^6}-2 x^3\right )^{2/3}}{4 \sqrt [3]{\sqrt {5} \sqrt {x^6}-x^3}} \\
\end{align*}