Internal
problem
ID
[1719]
Book
:
Elementary
differential
equations
with
boundary
value
problems.
William
F.
Trench.
Brooks/Cole
2001
Section
:
Chapter
2,
First
order
equations.
Exact
equations.
Integrating
factors.
Section
2.6
Page
91
Problem
number
:
9
Date
solved
:
Monday, January 27, 2025 at 05:32:28 AM
CAS
classification
:
[_rational, [_Abel, `2nd type`, `class B`]]
\begin{align*} 6 x y^{2}+2 y+\left (12 x^{2} y+6 x +3\right ) y^{\prime }&=0 \end{align*}
Time used: 0.004 (sec). Leaf size: 24
Time used: 61.558 (sec). Leaf size: 1544
\begin{align*}
y(x)\to -\frac {x^2 \left (-\sqrt {-\frac {48 \sqrt [3]{2} c_1}{\sqrt [3]{\sqrt {c_1{}^2 \left ((2 x+1)^4+256 c_1 x^6\right )}-c_1 (2 x+1)^2}}+\frac {(2 x+1)^2}{x^4}+\frac {6\ 2^{2/3} \sqrt [3]{\sqrt {c_1{}^2 \left ((2 x+1)^4+256 c_1 x^6\right )}-c_1 (2 x+1)^2}}{x^2}}\right )+\sqrt {2} x^2 \sqrt {\frac {24 \sqrt [3]{2} c_1}{\sqrt [3]{\sqrt {c_1{}^2 \left ((2 x+1)^4+256 c_1 x^6\right )}-c_1 (2 x+1)^2}}+\frac {(2 x+1)^2}{x^4}-\frac {3\ 2^{2/3} \sqrt [3]{\sqrt {c_1{}^2 \left ((2 x+1)^4+256 c_1 x^6\right )}-c_1 (2 x+1)^2}}{x^2}-\frac {(2 x+1)^3}{x^6 \sqrt {-\frac {48 \sqrt [3]{2} c_1}{\sqrt [3]{\sqrt {c_1{}^2 \left ((2 x+1)^4+256 c_1 x^6\right )}-c_1 (2 x+1)^2}}+\frac {(2 x+1)^2}{x^4}+\frac {6\ 2^{2/3} \sqrt [3]{\sqrt {c_1{}^2 \left ((2 x+1)^4+256 c_1 x^6\right )}-c_1 (2 x+1)^2}}{x^2}}}}+2 x+1}{12 x^2} \\
y(x)\to \frac {x^2 \sqrt {-\frac {48 \sqrt [3]{2} c_1}{\sqrt [3]{\sqrt {c_1{}^2 \left ((2 x+1)^4+256 c_1 x^6\right )}-c_1 (2 x+1)^2}}+\frac {(2 x+1)^2}{x^4}+\frac {6\ 2^{2/3} \sqrt [3]{\sqrt {c_1{}^2 \left ((2 x+1)^4+256 c_1 x^6\right )}-c_1 (2 x+1)^2}}{x^2}}+\sqrt {2} x^2 \sqrt {\frac {24 \sqrt [3]{2} c_1}{\sqrt [3]{\sqrt {c_1{}^2 \left ((2 x+1)^4+256 c_1 x^6\right )}-c_1 (2 x+1)^2}}+\frac {(2 x+1)^2}{x^4}-\frac {3\ 2^{2/3} \sqrt [3]{\sqrt {c_1{}^2 \left ((2 x+1)^4+256 c_1 x^6\right )}-c_1 (2 x+1)^2}}{x^2}-\frac {(2 x+1)^3}{x^6 \sqrt {-\frac {48 \sqrt [3]{2} c_1}{\sqrt [3]{\sqrt {c_1{}^2 \left ((2 x+1)^4+256 c_1 x^6\right )}-c_1 (2 x+1)^2}}+\frac {(2 x+1)^2}{x^4}+\frac {6\ 2^{2/3} \sqrt [3]{\sqrt {c_1{}^2 \left ((2 x+1)^4+256 c_1 x^6\right )}-c_1 (2 x+1)^2}}{x^2}}}}-2 x-1}{12 x^2} \\
y(x)\to \frac {1}{12} \left (-\frac {2 x+1}{x^2}-\sqrt {-\frac {48 \sqrt [3]{2} c_1}{\sqrt [3]{\sqrt {c_1{}^2 \left ((2 x+1)^4+256 c_1 x^6\right )}-c_1 (2 x+1)^2}}+\frac {(2 x+1)^2}{x^4}+\frac {6\ 2^{2/3} \sqrt [3]{\sqrt {c_1{}^2 \left ((2 x+1)^4+256 c_1 x^6\right )}-c_1 (2 x+1)^2}}{x^2}}-\sqrt {2} \sqrt {\frac {24 \sqrt [3]{2} c_1}{\sqrt [3]{\sqrt {c_1{}^2 \left ((2 x+1)^4+256 c_1 x^6\right )}-c_1 (2 x+1)^2}}+\frac {(2 x+1)^2}{x^4}-\frac {3\ 2^{2/3} \sqrt [3]{\sqrt {c_1{}^2 \left ((2 x+1)^4+256 c_1 x^6\right )}-c_1 (2 x+1)^2}}{x^2}+\frac {(2 x+1)^3}{x^6 \sqrt {-\frac {48 \sqrt [3]{2} c_1}{\sqrt [3]{\sqrt {c_1{}^2 \left ((2 x+1)^4+256 c_1 x^6\right )}-c_1 (2 x+1)^2}}+\frac {(2 x+1)^2}{x^4}+\frac {6\ 2^{2/3} \sqrt [3]{\sqrt {c_1{}^2 \left ((2 x+1)^4+256 c_1 x^6\right )}-c_1 (2 x+1)^2}}{x^2}}}}\right ) \\
y(x)\to -\frac {x^2 \sqrt {-\frac {48 \sqrt [3]{2} c_1}{\sqrt [3]{\sqrt {c_1{}^2 \left ((2 x+1)^4+256 c_1 x^6\right )}-c_1 (2 x+1)^2}}+\frac {(2 x+1)^2}{x^4}+\frac {6\ 2^{2/3} \sqrt [3]{\sqrt {c_1{}^2 \left ((2 x+1)^4+256 c_1 x^6\right )}-c_1 (2 x+1)^2}}{x^2}}-\sqrt {2} x^2 \sqrt {\frac {24 \sqrt [3]{2} c_1}{\sqrt [3]{\sqrt {c_1{}^2 \left ((2 x+1)^4+256 c_1 x^6\right )}-c_1 (2 x+1)^2}}+\frac {(2 x+1)^2}{x^4}-\frac {3\ 2^{2/3} \sqrt [3]{\sqrt {c_1{}^2 \left ((2 x+1)^4+256 c_1 x^6\right )}-c_1 (2 x+1)^2}}{x^2}+\frac {(2 x+1)^3}{x^6 \sqrt {-\frac {48 \sqrt [3]{2} c_1}{\sqrt [3]{\sqrt {c_1{}^2 \left ((2 x+1)^4+256 c_1 x^6\right )}-c_1 (2 x+1)^2}}+\frac {(2 x+1)^2}{x^4}+\frac {6\ 2^{2/3} \sqrt [3]{\sqrt {c_1{}^2 \left ((2 x+1)^4+256 c_1 x^6\right )}-c_1 (2 x+1)^2}}{x^2}}}}+2 x+1}{12 x^2} \\
\end{align*}